1. Introduction
Sober spaces were introduced by the Grothendieck school (Artin et al., Reference Artin, Grothendieck and Verdier1972, IV-4.2.1) and independently by Blanksma (Reference Blanksma1968). The importance of sober spaces is that they are precisely those spaces, which can be recovered from their lattice of open sets (cf. (Hofmann and Lawson, Reference Hofmann and Lawson1978, Proposition 2.7)).
In domain theory and non-Hausdorff topology, we encounter numerous links between topology and order theory (cf. Abramsky and Jung (Reference Abramsky and Jung1994); Gierz et al. (Reference Gierz, Hofmann, Keimel, Lawson, Mislove and Scott2003); Goubault (Reference Goubault-Larrecq2013)). The Scott topology
$\sigma (P)$
on a poset
$P$
is probably the most important one among the intrinsic
$T_0$
-topologies on
$P$
, and the sobriety is probably the most important and useful property of
$T_0$
-spaces. With the development of domain theory, another two properties also emerged as the very useful and important properties for non-Hausdorff topology: the property of being a
$d$
-space and the well-filteredness (see Abramsky and Jung (Reference Abramsky and Jung1994); Gierz et al. (Reference Gierz, Hofmann, Keimel, Lawson, Mislove and Scott2003); Goubault (Reference Goubault-Larrecq2013); Xu and Zhao (Reference Xu and Zhao2021)).
Rudin’s Lemma plays a crucial role in domain theory (see Abramsky and Jung (Reference Abramsky and Jung1994); Gierz et al. (Reference Gierz, Hofmann, Keimel, Lawson, Mislove and Scott2003); Goubault (Reference Goubault-Larrecq2013)). Heckmann and Keimel (Reference Heckmann and Keimel2013) provided a topological variant of Rudin’s Lemma. Based on the topological Rudin Lemma, Shen et al. (Reference Shen, Xi, Xu and Zhao2019) and Xu et al. (Reference Xu, Shen, Xi and Zhao2020b) introduced and investigated two new classes of subsets in
$T_0$
-spaces – Rudin sets and WD sets lying between the class of all closures of directed subsets and that of irreducible closed subsets. Using such subsets, Xu et al. (Reference Xu, Shen, Xi and Zhao2020b) defined two new types of
$T_0$
-spaces – Rudin spaces and WD spaces. The Rudin sets, WD sets, Rudin spaces, and WD spaces have been becoming important research objects of domain theory and non-Hausdorff topology (see Shen et al. (Reference Shen, Xi, Xu and Zhao2019); Liu et al. (Reference Liu, Li and Wu2020); Hou et al. (Reference Hou, Miao and Li2024); Xu et al. (Reference Xu, Shen, Xi and Zhao2020a,b, Reference Xu, Shen, Xi and Zhao2021, Reference Xu, Wen and Xi2023, Reference Xu, Miao and Li2025); Xu and Zhao (Reference Xu and Zhao2021); Zhang and Zhao (Reference Zhang and Zhao2023)).
In some situations, we only need the countable versions of Rudin sets and WD sets. Therefore, in Xu et al. (Reference Xu, Shen, Xi and Zhao2020a), the
$\omega$
-Rudin sets,
$\omega$
-WD sets,
$\omega$
-Rudin spaces, and
$\omega$
-WD spaces were introduced and studied. Using (
$\omega$
-)Rudin sets and (
$\omega$
-)
$\mathsf{WD}$
sets, some characterizations of (
$\omega$
-)well-filtered spaces and two decomposition theorems for the sobriety were obtained in Xu et al., (Reference Xu, Shen, Xi and Zhao2020a,b). Xu et al. (Reference Xu and Zhao2021) showed that every
$T_0$
-space with a first-countable sobrification is an
$\omega$
-Rudin space, and hence, every
$\omega$
-well-filtered space with a first-countable sobrification is sober. In particular, every second-countable
$\omega$
-well-filtered space is sober. Furthermore, it was proved in Xu et al. (Reference Xu, Miao and Li2025) that if the sobrification (especially, the Hoare power space) of a
$T_0$
-space
$X$
is a Fréchet space, then
$X$
is an
$\omega$
-Rudin space, whence every
$\omega$
-well-filtered space for which its sobrification (especially, its Hoare power space) is a Fréchet space is sober.
The main purpose of this paper is to characterize
$\omega$
-Rudin sets and
$\omega$
-Rudin spaces via sequence convergence and give some important applications of such characterizations. The paper is organized as follows.
Section 2 provides some fundamental definitions and notations, and also a few basic results related with irreducible sets, sober spaces, well-filtered spaces, and the Scott topology.
In Section 3, we briefly recall the concepts and some results of (
$\omega$
-)well-filtered spaces, (
$\omega$
-)Rudin sets, and (
$\omega$
-)Rudin spaces. We show that every countable locally hypercompact
$T_0$
-space is an
$\omega$
-Rudin space. It remains open whether a countable locally compact
$T_0$
-space is an
$\omega$
-Rudin space.
In Section 4, we give some characterizations of
$\omega$
-Rudin sets and
$\omega$
-Rudin spaces via sequence convergence. For an irreducible closed set
$A$
of a
$T_0$
-space
$X$
, we prove that the following four conditions are equivalent: (1)
$A$
is an
$\omega$
-Rudin set; (2) there is
$\{a_n : n\in \mathbb{N}\}\subseteq A$
such that the sequence
$(a_n)_{n\in \mathbb{N}}$
simultaneously converges to all points of
$A$
; (3) there is
$\{a_n : n\in \mathbb{N}\}\subseteq A$
such that the sequence
$(\overline {\{a_n\}})_{n\in \mathbb{N}}$
converges to
$A$
in the Hoare power space of
$X$
; (4) there is
$\{a_n : n\in \mathbb{N}\}\subseteq A$
such that the sequence
$(\overline {\{a_n\}})_{n\in \mathbb{N}}$
converges to
$A$
in the sobrification of
$X$
. Based on these characterizations, we obtain some characterizations of
$\omega$
-Rudin spaces and sober spaces. In particular, we show that for a complete lattice
$L$
, its Scott space
$\Sigma L$
is sober iff for any nonempty Scott irreducible closed set
$A$
of
$L$
, there is
$\{a_n : n\in \mathbb{N}\}\subseteq A$
such that the sequence
$(a_n)_{n\in \mathbb{N}}$
simultaneously Scott converges to all points of
$A$
or, equivalently, the sequence
$(\overline {\{a_n\}})_{n\in \mathbb{N}}$
converges to
$A$
in the sobrification of
$\Sigma L$
. Several related examples are presented.
In Section 5, we investigate some basic properties of
$\omega$
-Rudin spaces. It is proved that the property of being an
$\omega$
-Rudin space is retractive, productive, and closed-hereditary. We give an example to show that it is not saturated-hereditary.
Section 6 is mainly devoted to discuss some properties of Fréchet spaces and first-countable spaces. We show that if the sobrification (especially, the Hoare power space) of a
$T_0$
-space
$X$
is a Fréchet space, then
$X$
is an
$\omega$
-Rudin space. Hence, every
$\omega$
-well-filtered space for which its sobrification (especially, its Hoare power space) is a Fréchet space is sober. Two examples are provided to show that a Fréchet well-filtered space may not be sober, a first-countable
$T_0$
-space may not be an
$\omega$
-Rudin space, and a first-countable
$\omega$
-well-filtered space may not be sober.
In Section 7, we prove that the category
$\boldsymbol{\omega }$
-
$\mathbf{Rud}$
of
$\omega$
-Rudin spaces does not have equalizers, and hence,
$\boldsymbol{\omega }$
-
$\mathbf{Rud}$
is not reflective in the category
$\mathbf{Top}_{0}$
of all
$T_0$
-spaces. As an explicit example, we show that the
$\boldsymbol{\omega }$
-
$\mathbf{Rud}$
-reflection of
$X_{coc}$
does not exist, where
$X_{coc}$
is an uncountable set endowed with the co-countable topology (the empty set and the complements of countable subsets of
$X$
are open). As a byproduct, we get the non-reflectivity of three related categories in
$\mathbf{Top}_0$
.
In Section 8, we discuss the Smyth power spaces of
$\omega$
-Rudin spaces. It is shown that if the Smyth power space of a
$T_0$
-space
$X$
is an
$\omega$
-Rudin space, then
$X$
is an
$\omega$
-Rudin space, and if
$X$
is an
$\omega$
-well-filtered
$\omega$
-Rudin space, then its Smyth power space is a sober space and hence an
$\omega$
-Rudin space. Finally, we pose a question whether the Smyth power space of an
$\omega$
-Rudin space is still an
$\omega$
-Rudin space.
2. Preliminaries
In this section, we briefly recall some fundamental concepts and notations that will be used in the paper. A few basic results related with irreducible sets, sober spaces, well-filtered spaces, and the Scott topology are listed. For further details, we refer the reader to Abramsky and Jung (Reference Abramsky and Jung1994); Gierz et al. (Reference Gierz, Hofmann, Keimel, Lawson, Mislove and Scott2003), Goubault (Reference Goubault-Larrecq2013), Schalk (Reference Schalk1993).
For a set
$X$
, let
$|X|$
be the cardinality of
$X$
and
$2^X$
the set of all subsets of
$X$
. The set of all natural numbers is denoted by
$\mathbb{N}$
. Let
$\mathbb{N}^{+}=\mathbb{N}\setminus \{0\}$
. When
$\mathbb{N}$
is regarded as a poset (in fact, a chain), the order on
$\mathbb{N}$
is the usual order of natural numbers. Let
$\omega =|\mathbb{N}|$
and
$\omega _1$
denote the first uncountable ordinal.
Giving a poset
$P$
and
$A\subseteq P$
, we define
$\mathord {\downarrow }A=\{x\in P: x\leq a \mbox{for some } a\in A\}$
and
$\mathord {\uparrow }A=\{x\in P: x\geq a \mbox{ for some } a\in A\}$
. For
$x\in P$
, we write
$\mathord {\downarrow }x$
for
$\mathord {\downarrow }\{x\}$
and
$\mathord {\uparrow }x$
for
$\mathord {\uparrow }\{x\}$
. A subset
$A$
is called a lower set (resp., an upper set) if
$A=\mathord {\downarrow }A$
(resp.,
$A=\mathord {\uparrow }A$
). Let
$P^{(\lt \omega )}=\{F\subseteq P : F \mbox{is a finite set}\}$
and
$\mathbf{Fin}\,\! P=\{\uparrow \! F : F\in P^{(\lt \omega )}\setminus \{\emptyset \}\}$
. The set of all countable subsets of
$P$
is denoted by
$P^{(\leq \omega )}$
. For a nonempty subset
$A$
of
$P$
, define
$\operatorname{min} (A)=\{a\in A : a$
is a minimal element of
$A\}$
.
A nonempty subset
$D$
of a poset
$P$
is directed if every two elements in
$D$
have an upper bound in
$D$
. The set of all directed sets of
$P$
is denoted by
$\mathscr D(P)$
. The poset
$P$
is called a directed complete poset, or dcpo for short, provided that
$\vee D$
exists in
$P$
for any
$D\in \mathscr D(P)$
.
$I\subseteq P$
is called an ideal of
$P$
if
$I$
is a directed lower subset of
$P$
. Let
$\operatorname{Id} (P)$
be the poset (with the order of set inclusion) of all ideals of
$P$
.
A subset
$U$
of a poset
$Q$
is Scott open if (i)
$U=\mathord {\uparrow }U$
, and (ii) for any directed subset
$D$
of
$Q$
with
$\vee D$
existing,
$\vee D\in U$
implies
$D\cap U\neq \emptyset$
. All Scott open subsets of
$Q$
form a topology. This topology is called the Scott topology on
$Q$
and denoted by
$\sigma (Q)$
. The space
$\Sigma \,\! Q=(Q,\sigma (Q))$
is called the Scott space of
$Q$
. The upper topology
$\upsilon (P)$
on
$P$
is the topology that has
$\{P\setminus \downarrow x : x\in P\}$
as a subbase. The upper sets of
$Q$
form the (upper) Alexandroff topology
$\alpha (Q)$
.
For a
$T_0$
-space
$X$
, we use
$\leq _X$
to represent the specialization order of
$X$
, that is,
$x\leq _X y$
iff
$x\in \overline {\{y\}}$
. In the following, when a
$T_0$
-space
$X$
is considered as a poset, the order always refers to the specialization order if no other explanation is given. Let
$\mathscr O(X)$
(resp.,
$\Gamma (X)$
) be the set of all open subsets (resp., closed subsets) of
$X$
. Define
$\mathscr S_c(X)=\{\overline {{\{x\}}} : x\in X\}$
and
$\mathscr D_c(X)=\{\overline {D} : D\in \mathscr D(X)\}$
.
For the following definition and related conceptions, please refer to Gierz et al. (Reference Gierz, Hofmann, Keimel, Lawson, Mislove and Scott2003), Goubault (Reference Goubault-Larrecq2013).
Definition 1.
Let
$X$
be a
$T_0$
-space.
-
(i) The space
$X$
is called a
$d$
-space (or monotone convergence space) if
$X$
(with the specialization order) is a dcpo and
$\mathscr O(X) \subseteq \sigma (X)$
(cf. Wyler (Reference Wyler1981); Gierz et al. (Reference Gierz, Hofmann, Keimel, Lawson, Mislove and Scott2003)).
-
(ii) A nonempty subset
$A$
of
$X$
is irreducible if for any
$F_1, F_2\in \Gamma (X)$
,
$A \subseteq F_1\cup F_2$
implies
$A \subseteq F_1$
or
$A \subseteq F_2$
. Denote by
${\mathsf{Irr}}(X)$
(resp.,
${\mathsf{Irr}}_c(X)$
) the set of all irreducible (resp., irreducible closed) subsets of
$X$
. -
(iii) The space
$X$
is called sober, if for any
$A\in {\mathsf{Irr}}_c(X)$
, there is a (unique) point
$x\in X$
such that
$A=\overline {\{x\}}$
. The category of all sober spaces and continuous mappings is denoted by
$\mathbf{Sob}$
. -
(iv) A subset
$B$
of
$X$
is called saturated if
$B$
equals the intersection of all open sets containing it (equivalently,
$B$
is an upper set in the specialization order). Let
$\mathsf{K}(X)$
be the poset of all nonempty compact saturated subsets of
$X$
endowed with the Smyth preorder
$\sqsubseteq$
, that is, for
$K_1,K_2\in \mathord {\mathsf{K}}(X)$
,
$K_1\sqsubseteq K_2$
iff
$K_2\subseteq K_1$
.
$(\mathsf{K}(X), \sqsubseteq )$
is called the Smyth power poset of
$X$
. Let
$ \mathscr S\,^{u}(X)=\{\uparrow x : x\in X\}$
. -
(v) The space
$X$
is called well-filtered if for any filtered family
$\mathscr{K}\subseteq \mathord {\mathsf{K}}(X)$
and any open set
$U$
,
$\bigcap \mathscr{K}{\subseteq } U$
implies
$K{\subseteq } U$
for some
$K{\in }\mathscr{K}$
. The category of all well-filtered spaces and continuous mappings is denoted by
$\mathbf{Top}_w$
.
Clearly, every subset of a
$T_0$
-space
$X$
that is directed under
$\leq _X$
is irreducible and
$\Sigma \,\! P$
is a
$d$
-space for any dcpo
$P$
.
The following two lemmas on irreducible sets are well-known.
Lemma 2.
Let
$X$
be a space and
$Y$
a subspace of
$X$
. Then, the following conditions are equivalent for a subset
$A\subseteq Y$
:
-
(1)
$A$
is an irreducible subset of
$Y$
. -
(2)
$A$
is an irreducible subset of
$X$
. -
(3)
$\textrm {cl}_X A$
is an irreducible closed subset of
$X$
.
Lemma 3.
If
$f : X \longrightarrow Y$
is continuous and
$A\in {\mathsf{Irr}} (X)$
, then
$f(A)\in {\mathsf{Irr}} (Y)$
.
Lemma 4 (Xu et al. Reference Xu, Shen, Xi and Zhao2020b, Corollary 2.7). Let
$X=\prod _{i\in I}X_i$
be the product space of
$T_0$
-spaces
$X_i$
$(i\in I)$
. If
$A\in {\mathsf{Irr}}_c(X)$
, then
$A=\prod _{i\in I}p_i(A)$
and
$p_i(A)\in {\mathsf{Irr}}_c (X_i)$
for each
$i\in I$
.
The following result is well-known (see, e.g., Hoffmann (Reference Hoffmann1979); Heckmann and Keimel (Reference Heckmann and Keimel2013)).
Lemma 5.
Let
$P$
be a poset. Then,
-
(1)
${\mathsf{Irr}} ((P, \alpha (P)))=\mathscr D(P)$
and
${\mathsf{Irr}}_c((P, \alpha (P)))=\mathscr{D}_c((P, \alpha (P)))=\operatorname{Id}(P)$
. -
(2)
$\mathsf{K}((P, \alpha (P)))=\mathbf{Fin}\,\!P$
.
As in Erné (Reference Erné2018), a topological space
$X$
is locally hypercompact if for each
$x\in X$
and each open neighborhood
$U$
of
$x$
, there is
$\mathord {\uparrow } F\in \mathbf{Fin}\,\! X$
such that
$x\in \textrm {int}\,\mathord {\uparrow } F\subseteq \mathord {\uparrow } F\subseteq U$
.
Lemma 6 (Erné Reference Erné2018, Proposition 3.2). Let
$X$
be a locally hypercompact
$T_0$
-space and
$A\in {\mathsf{Irr}}(X)$
. Then, there exists a directed subset
$D\subseteq \mathord {\downarrow } A$
such that
$\overline {A}=\overline {D}$
.
The following implications are well-known (see, e.g., Xu et al. (Reference Xu, Shen, Xi and Zhao2020b)):
\begin{equation*} \textrm{sober spaces} \Rightarrow \textrm{well-filtered spaces} \Rightarrow d\textrm{-spaces}. \end{equation*}
A topological property
$S$
is said to be hereditary (saturated-hereditary, closed-hereditary, open-hereditary etc.) if for any space
$X$
that has the property
$S$
, every subspace (every saturated subspace, closed subspace, open subspace etc.) of
$X$
also has the property
$S$
. The property
$S$
is said to be productive if the product
$\prod _{i\in I}X_i$
has the property
$S$
whenever all spaces
$X_i$
have this property.
Proposition 7. The well-filteredness is both closed-hereditary and saturated-hereditary.
Proof.
Suppose that
$X$
is a well-filtered space and
$A$
is a nonempty closed subspace of
$X$
. Assume that
$\{K_i : i\in I\}\subseteq \mathord {\mathsf{K}} (A)$
is a filtered family and
$V\in \mathscr O(A)$
with
$\bigcap _{i\in I}K_i \subseteq V$
. Then,
$V=U_V\cap A$
for some
$U_V\in \mathscr O(X)$
. Clearly,
$\{\uparrow _X K_i : i\in I\}\subseteq \mathord {\mathsf{K}} (X)$
is a filtered family. For each
$i\in I$
, by
$K_i=\uparrow _A K_i$
we have
$(\uparrow _X K_i)\cap A=K_i$
, and hence
$\bigcap _{i\in I}\uparrow _X K_i \subseteq V\cup (X\setminus A)\subseteq U_V\cup (X\setminus A)$
. By the well-filteredness of
$X$
, there is
$i\in I$
such that
$\uparrow _X K_i\subseteq U_V\cup (X\setminus A)$
(or equivalently,
$A\cap \mathord {\uparrow }_X K_i\subseteq U_V$
), whence
$K_i\subseteq U_V\cap A=V$
. Thus,
$A$
is well-filtered.
Now, let
$S$
be a nonempty saturated subspace of
$X$
. We show that
$S$
is well-filtered. Assume that
$\{G_j : j\in J\}\subseteq \mathord {\mathsf{K}} (S)$
is a filtered family and
$W\in \mathscr O(S)$
with
$\bigcap _{j\in J}G_j \subseteq W$
. Then,
$W=U_W\cap S$
for some
$U_W\in \mathscr O(X)$
. As
$S$
is saturated, we have
$S=\uparrow _X S$
and hence
$G_j=\uparrow _S G_j=\uparrow _X G_j$
for all
$j\in J$
. It follows that
$\{G_j : j\in J\}\subseteq \mathord {\mathsf{K}} (X)$
and
$\bigcap _{j\in J}G_j \subseteq U_W$
. By the well-filteredness of
$X$
, there is
$j\in J$
such that
$G_j\subseteq U_W$
, whence
$G_j\subseteq U_W\cap S=W$
. Thus,
$S$
is well-filtered.
A
$T_0$
-space
$X$
is called core-compact if
$\mathscr O(X)$
is a continuous lattice (see (Goubault Reference Goubault-Larrecq2013, Definition 5.1.5 and Definition 5.2.3)). Clearly, every locally compact space is core-compact (see, e.g., (Gierz et al. Reference Gierz, Hofmann, Keimel, Lawson, Mislove and Scott2003, Examples I-1.7)).
Using continuous lattice techniques, Hofmann and Lawson (Reference Hofmann and Lawson1978) showed that every sober core-compact space is locally compact (see also (Gierz et al. Reference Gierz, Hofmann, Keimel, Lawson, Mislove and Scott2003, Theorem V-5.6)).
Proposition 8 (Hofmann and Lawson Reference Hofmann and Lawson1978, Theorem 4.5 or Corollary 4.6). Let
$X$
be a sober space. Then,
$X$
is locally compact iff it is core-compact.
Proposition 8 has been improved in Kou (Reference Kou2001); Lawson et al. (Reference Lawson, Wu and Xi2020) and Xu et al. (Reference Xu, Shen, Xi and Zhao2020b) as follows.
Theorem 9 (Kou, Reference Kou2001, Theorem 2.3) (Lawson et al., Reference Lawson, Wu and Xi2020, Theorem 3.1) (Xu et al. Reference Xu, Shen, Xi and Zhao2020b, Theorem 6.16). For a
$T_0$
-space
$X$
, the following conditions are equivalent:
-
(1)
$X$
is locally compact and sober.
-
(2)
$X$
is locally compact and well-filtered.
-
(3)
$X$
is core-compact and sober.
-
(4)
$X$
is core-compact and well-filtered.
Hofmann and Lawson (Reference Hofmann and Lawson1978) (see also (Gierz et al. Reference Gierz, Hofmann, Keimel, Lawson, Mislove and Scott2003, Exercise V-5.25)) gave a second-countable core-compact
$T_0$
-space
$X$
in which every compact subset of
$X$
has empty interior, and hence, it is not locally compact.
In Xi and Lawson (Reference Xi and Lawson2017), the following useful result was given.
Proposition 10 (Xi and Lawson Reference Xi and Lawson2017, Corollary 3.2). For a complete lattice
$L$
,
$\Sigma L$
is well-filtered.
For any topological space
$X$
,
$\mathscr G\subseteq 2^{X}$
and
$A\subseteq X$
, let
$\Diamond _{\mathscr G} A=\{G\in \mathscr G : G\cap A\neq \emptyset \}$
and
$\Box _{\mathscr G} A=\{G\in \mathscr G : G\subseteq A\}$
. The symbols
$\Diamond _{\mathscr G} A$
and
$\Box _{\mathscr G} A$
will be simply written as
$\Diamond A$
and
$\Box A$
, respectively, if there is no ambiguous occurrence.
The lower Vietoris topology on
$\mathscr{G}$
is the topology that has
$\{\Diamond U : U\in \mathscr O(X)\}$
as a subbase, and the resulting space is denoted by
$P_H(\mathscr{G}\,)$
. If
$\mathscr{G}\subseteq {\mathsf{Irr}} (X)$
, then
$\{\Diamond _{\mathscr{G}} U : U\in \mathscr O(X)\}$
is a topology on
$\mathscr{G}$
. The space
$P_H(\Gamma (X)\setminus \{\emptyset \})$
is called the Hoare power space or lower space of
$X$
and is denoted by
$P_H(X)$
for short (cf. Schalk (Reference Schalk1993)). Clearly,
$P_H(X)=(\Gamma (X)\setminus \{\emptyset \}, \upsilon (\Gamma (X)\setminus \{\emptyset \}))$
. So
$P_H(X)$
is always sober (see, e.g., (Zhao and Ho Reference Zhao and Ho2015, Corollary 4.10) or (Xu et al. Reference Xu, Shen, Xi and Zhao2020b, Proposition 2.9)).
The upper Vietoris topology on
$\mathscr{G}$
is the topology that has
$\{\Box _{\mathscr{G}} U : U\in \mathscr O(X)\}$
as a base, and the resulting space is denoted by
$P_S(\mathscr{G}\,)$
. The space
$P_S(\mathord {\mathsf{K}}(X))$
, denoted shortly by
$P_S(X)$
, is called the Smyth power space or upper space of
$X$
(cf. Heckmann (Reference Heckmann1992); Schalk (Reference Schalk1993)).
Remark 11. Let
$X$
be a
$T_0$
-space.
-
(1) If
$\mathscr{S}_c(X)\subseteq \mathscr{G}\subseteq 2^{X}$
, then the specialization order on
$P_H(\mathscr{G}\,)$
is the set inclusion order, and the canonical mapping
$\eta _{X}: X\longrightarrow P_H(\mathscr{G}\,)$
, given by
$\eta _X(x)=\overline {\{x\}}$
, is a topological embedding (cf. Schalk (Reference Schalk1993); Gierz et al. (Reference Gierz, Hofmann, Keimel, Lawson, Mislove and Scott2003)). -
(2) The space
$X^s=P_H({\mathsf{Irr}}_c(X))$
with the canonical mapping
$\eta _{X}: X\longrightarrow X^s$
is the sobrification of
$X$
(cf. Gierz et al. (Reference Gierz, Hofmann, Keimel, Lawson, Mislove and Scott2003); Goubault (2013)).
Remark 12. Let
$X$
be a
$T_0$
-space. Then,
-
(1) The specialization order on
$P_S(X)$
is the Smyth order (i.e.,,
$\leq _{P_S(X)}=\sqsubseteq$
). -
(2) The canonical mapping
$\xi _X: X\longrightarrow P_S(X)$
,
$x\mapsto \mathord {\uparrow } x$
, is a topological embedding (cf. Heckmann (Reference Heckmann1992); Schalk (Reference Schalk1993); Heckmann and Keimel (Reference Heckmann and Keimel2013)). -
(3)
$X$
is homeomorphic to the subspace
$\mathscr S^{\,u}(X)$
of
$P_S(X)$
by means of
$\xi _X$
.
Lemma 13.
For a
$T_0$
-space
$X$
, the following conditions are equivalent:
-
(1)
$X$
is second-countable.
-
(2)
$P_S(X)$
is second-countable.
-
(3)
$P_H(X)$
is second-countable.
-
(4)
$X^s$
is second-countable.
Proof.
Since the second-countability is a hereditary property, by Remark 11 and Remark 12, we have the following implications: (2)
$\Rightarrow$
(1), (3)
$\Rightarrow$
(1) and (4)
$\Rightarrow$
(1).
(1)
$\Rightarrow$
(2): This implication was indicated in the proof of (Brecht and Kawai Reference Brecht and Kawai2019, Proposition 6.16).
(3)
$\Rightarrow$
(1): Let
$\mathscr{B}\subseteq \mathscr O(X)$
be a countable base of
$X$
and let
$\mathscr{B}_H=\{\Diamond _{\Gamma (X)\setminus \{\emptyset \}}U : U\in \mathscr B\}$
. Then, it is easy to verify that
$\mathscr{B}_H$
is a countable base of
$P_H(X)$
.
(4)
$\Rightarrow$
(1): Let
$\{U_n : n\in \mathbb{N}\}\subseteq \mathscr O(X)$
be a countable base of
$X$
. Then,
$\{\Diamond _{\operatorname{Irr}_c(X)}U_n : n\in \mathbb{N}\}$
is a countable base of
$X^s$
.
For two topological spaces
$X$
and
$Y$
,
$Y$
is said to be a retract of
$X$
if there are two continuous mappings
$f : X\rightarrow Y$
and
$g : Y\rightarrow X$
such that
$f\circ g=id_{Y}$
.
A topological property
$S$
is said to be retractive if for any space
$X$
that has the property
$S$
, every retract of
$X$
also has the property
$S$
.
We have the following folklore result, which can be easily verified.
Lemma 14.
Let
$\{X_i:i\in I\}$
be a family of
$T_0$
-spaces. Then, for any
$j\in I$
,
$X_j$
is a retract of the product space
$\prod _{i\in I}X_i$
.
3. Topological Rudin Lemma and Rudin Sets
Rudin’s Lemma is a useful tool in topology and plays a crucial role in domain theory (see Gierz et al., (Reference Gierz, Lawson and Stralka1983, Reference Gierz, Hofmann, Keimel, Lawson, Mislove and Scott2003); Abramsky and Jung (Reference Abramsky and Jung1994); Goubault (Reference Goubault-Larrecq2013); Heckmann (Reference Heckmann1992)). Rudin (Reference Rudin1980) proved her lemma by transfinite methods, using the Axiom of Choice. Heckmann and Keimel (Reference Heckmann and Keimel2013) presented the following topological variant of Rudin’s Lemma.
Lemma 15 (Topological Rudin Lemma). Let
$X$
be a topological space and
$\mathscr{A}$
an irreducible subset of the Smyth power space
$P_S(X)$
. Then, every closed set
$C {\subseteq } X$
that meets all members of
$\mathscr{A}$
contains a minimal irreducible closed subset
$A$
that still meets all members of
$\mathscr{A}$
.
For a
$T_0$
-space
$X$
and
$\mathscr{K}\subseteq \mathord {\mathsf{K}}(X)$
, let
$M(\mathscr{K}\,)=\{A\in \Gamma (X) : K\cap A\neq \emptyset \mbox{for all} K\in \mathscr{K}\,\}$
(i.e.,
$\mathscr K\subseteq \Diamond A$
) and
$m(\mathscr{K}\,)=\{A\in \Gamma (X) : A \mbox{is a minimal member of} M(\mathscr{K}\,)\}$
.
Based on the topological Rudin Lemma, Shen et al. (Reference Shen, Xi, Xu and Zhao2019) and Xu et al. (Reference Xu, Shen, Xi and Zhao2020b) introduced and investigated two new classes of subsets in
$T_0$
-spaces – Rudin sets and WD sets lying between the class of all closures of directed subsets and that of irreducible closed subsets. Using such subsets, they defined two new types of topological spaces – Rudin spaces and WD spaces.
Definition 16. (Shen et al. Reference Shen, Xi, Xu and Zhao2019, Definition 2.1) (Xu et al. Reference Xu, Shen, Xi and Zhao2020b, Definition 4.6 and Definition 6.1). Let
$X$
be a
$T_0$
-space and
$A$
a nonempty closed subset of
$X$
.
-
(i)
$A$
is said to be a Rudin set, if there exists a filtered family
$\mathscr K\subseteq \mathord {\mathsf{K}}(X)$
such that
$A\in m(\mathscr K)$
(i.e.,,
$A$
is a minimal closed set that intersects all members of
$\mathscr K$
). Denote by
$\mathsf{RD} (X)$
the set of all Rudin sets of
$X$
. The space
$X$
is called a Rudin space, if
${\mathsf{Irr}}_c(X)=\mathsf{RD}(X)$
or, equivalently, all irreducible closed subsets of
$X$
are Rudin sets. The category of all Rudin spaces and continuous mappings is denoted by
$\mathbf{Rud}$
. -
(ii) A subset
$B$
of
$X$
is called a well-filtered determined set, a
$\mathord {\mathsf{WD}}$
set for short, if for any continuous mapping
$ f:X\longrightarrow Y$
to a well-filtered space
$Y$
, there exists a unique
$y_B\in Y$
such that
$\overline {f(B)}=\overline {\{y_B\}}$
. Denote by
$\mathsf{WD}(X)$
the set of all closed well-filtered determined subsets of
$X$
. The space
$X$
is called a well-filtered determined space, a
$\mathsf{WD}$
space for short, if all irreducible closed subsets of
$X$
are well-filtered determined, that is,
${\mathsf{Irr}}_c(X)=\mathsf{WD}(X)$
. The category of all
$\mathsf{WD}$
spaces and continuous mappings is denoted by
$\mathbf{WD}$
.
Lemma 17 (Xu et al. Reference Xu, Shen, Xi and Zhao2020b, Proposition 6.2). Let
$X$
be a
$T_0$
-space. Then,
$\mathscr{S}_c(X)\subseteq \mathscr{D}_c(X)\subseteq \mathsf{RD}(X)\subseteq \mathsf{WD}(X)\subseteq {\mathsf{Irr}}_c(X)$
.
By Lemma 5 and Lemma 17, we get the following result.
Proposition 18.
For any poset
$P$
,
$(P, \alpha (P))$
is a Rudin space.
Proposition 19 (Xu et al. Reference Xu, Shen, Xi and Zhao2020b, Theorem 6.10). Every locally compact
$T_0$
-space is a Rudin space.
In some situations, we only need the following countable version of well-filteredness and countable versions of Rudin sets and WD sets.
Definition 20 (Xu et al. Reference Xu, Shen, Xi and Zhao2020a, Definition 3.9). A
$T_0$
-space
$X$
is called
$\omega$
-well-filtered, if for any countable filtered family
$\{K_n:n\lt \omega \}\subseteq \mathord {\mathsf{K}} (X)$
and
$U\in \mathscr O(X)$
, it satisfies
\begin{equation*}\bigcap _{n\lt \omega }K_n\subseteq U \ \Rightarrow \ \exists n_0\lt \omega , K_{n_0}\subseteq U.\end{equation*}
The category of all
$\omega$
-well-filtered spaces and continuous mappings is denoted by
$\mathbf{Top}_{\omega \mbox{-}w}$
.
An
$\omega$
-filtered space may not be a well-filtered space (see Example 66 in Section 6).
Definition 21. (Xu et al. Reference Xu, Shen, Xi and Zhao2020a, Definition 5.1 and Definition 5.4). Let
$X$
be a
$T_0$
-space and
$A$
a nonempty closed subset of
$X$
.
-
(i)
$A$
is said to be an
$\omega$
-Rudin set, if there exists a countable filtered family
$\mathscr K\subseteq \mathord {\mathsf{K}}(X)$
such that
$A\in m(\mathscr K)$
. Denote by
$\omega$
-
$\mathsf{RD}(X)$
the set of all
$\omega$
-Rudin sets of
$X$
. The space
$X$
is called
$\omega$
-Rudin space, if
${\mathsf{Irr}}_c(X)=\omega$
-
$\mathsf{RD}(X)$
or, equivalently, all irreducible closed subsets of
$X$
are
$\omega$
-Rudin sets. The category of all
$\omega$
-Rudin spaces and continuous mappings is denoted by
$\boldsymbol{\omega }$
-
$\mathbf{Rud}$
. -
(ii) A subset
$B$
of
$X$
is called an
$\omega$
-well-filtered determined set, an
$\omega$
-
$\mathord {\mathsf{WD}}$
set for short, if for any continuous mapping
$ f:X\longrightarrow Y$
to an
$\omega$
-well-filtered space
$Y$
, there exists a unique
$y_B\in Y$
such that
$\overline {f(B)}=\overline {\{y_B\}}$
. Denote by
$\omega$
-
$\mathsf{WD}(X)$
the set of all closed
$\omega$
-well-filtered determined subsets of
$X$
. The space
$X$
is called an
$\omega$
-well-filtered determined space, an
$\omega$
-
$\mathsf{WD}$
space for short, if all irreducible closed subsets of
$X$
are
$\omega$
-well-filtered determined, that is,
${\mathsf{Irr}}_c(X)=\omega$
-
$\mathsf{WD}(X)$
. The category of all
$\omega$
-
$\mathsf{WD}$
spaces and continuous mappings is denoted by
$\boldsymbol{\omega }$
-
$\mathbf{WD}$
.
A Rudin space may not be an
$\omega$
-Rudin space (see Example 51 or Example 66 below).
For a poset
$P$
, let
$\mathscr D^\omega (P)=\{D\subseteq X : D \mbox{is countable and directed}\}$
. The poset
$P$
is called an
$\omega$
-dcpo, if for any
$D\in \mathscr D^\omega (P)$
,
$\vee D$
exists. For a
$T_0$
-space
$X$
, let
$\mathscr D^\omega _c(X)=\{\overline {D} : D\in \mathscr D^\omega (X)\}$
.
For
$\omega$
-Rudin sets and
$\omega$
-
$\mathsf{WD}$
sets, we have the following result similar to Lemma 17.
Lemma 22 (Xu et al. Reference Xu, Shen, Xi and Zhao2020a, Proposition 5.5). Let
$X$
be a
$T_0$
-space. Then,
$\mathscr{S}_c(X)\subseteq \mathscr{D}_c^{\omega }(X)\subseteq \omega$
-
$\mathsf{RD}(X)\subseteq \omega$
-
$\mathsf{WD}(X)\subseteq {\mathsf{Irr}}_c(X)$
.
From Lemma 6 and Lemma 22, we deduce the following.
Proposition 23.
Every countable locally hypercompact
$T_0$
-space is an
$\omega$
-Rudin space.
But we do not know whether a countable locally compact
$T_0$
-space is an
$\omega$
-Rudin space (note that it is a Rudin space by Proposition 19), that is, we have the following question.
Question 24.
Is every countable locally compact
$T_0$
-space an
$\omega$
-Rudin space?
Lemma 25 (Abramsky and Jung Reference Abramsky and Jung1994, Proposition 2.2.13) (Xu et al. Reference Xu, Miao and Li2025, Lemma 2.1). Let
$P$
be a poset and
$D\in \mathscr D^\omega (P)$
. Then, there exists a countable chain
$C\subseteq D$
such that
$\mathord {\downarrow } D=\mathord {\downarrow } C$
. Hence,
$\vee C$
exists and
$\vee C=\vee D$
whenever
$\vee D$
exists. If
$D$
has no largest element, then
$C$
can be chosen to be a strictly ascending chain.
From Lemma 25, we directly get the following result.
Proposition 26.
A
$T_0$
-space
$X$
is
$\omega$
-well-filtered iff for any countable descending chain
$K_0\supseteq K_1\supseteq K_2\supseteq \ldots \supseteq K_n\supseteq \ldots$
of compact saturated subsets of
$X$
and
$U\in \mathscr O(X)$
, the following implication holds:
\begin{equation*}\bigcap _{n\lt \omega }K_n\subseteq U\ \Rightarrow \ \exists n_0\lt \omega , \ K_{n_0}\subseteq U.\end{equation*}
Proposition 27.
Let
$X$
be a
$T_0$
-space and
$A$
a nonempty closed subset of
$X$
. Then, the following two conditions are equivalent:
-
(1)
$A$
is an
$\omega$
-Rudin set.
-
(2) There is a countable descending chain
$K_1\supseteq K_2\supseteq \ldots \supseteq K_n\supseteq \ldots$
of compact saturated subsets of
$X$
such that
$A$
is a minimal closed set that intersects all members
$K_n$
.
Proof.
(1)
$\Rightarrow$
(2): As
$A$
is an
$\omega$
-Rudin set, there exists a countable filtered family
$\mathscr K\subseteq \mathord {\mathsf{K}}(X)$
such that
$A\in m(\mathscr K)$
. By Lemma 25, there exists a countable chain
$K_1\supseteq K_2\supseteq \ldots \supseteq K_n\supseteq \ldots$
in
$\mathscr K$
such that
$\mathord {\downarrow }_{\mathsf{K}(X)} \mathscr K=\mathord {\downarrow }_{\mathord {\mathsf{K}}(X)}\{K_n : n\in \mathbb{N}^{+}\}$
(note that the order of
$\mathsf{K}(X)$
is the Smyth order). Then, for each
$n\in \mathbb{N}^{+}$
, there is
$K\in \mathscr K$
with
$K_n\sqsubseteq K$
(equivalently,
$K\subseteq K_n$
) and hence
$\emptyset \neq A\cap K\subseteq A\cap K_n$
. So
$A\in M(\{K_n : n\in \mathbb{N}^{+}\}$
. If
$B$
is a proper closed subset of
$A$
, then by
$A\in m(\mathscr K)$
there is
$G\in \mathscr K$
such that
$B\cap G=\emptyset$
. By
$G\in \mathscr K\subseteq \mathord {\downarrow }_{\mathord {\mathsf{K}}(X)}\{K_n : n\in \mathbb{N}^{+}\}$
, there is
$m\in \mathbb{N}^{+}$
such that
$K_m\subseteq G$
. Then,
$B\cap K_m=\emptyset$
. Thus,
$A$
is a minimal closed set that intersects all members
$K_n$
.
(2)
$\Rightarrow$
(1): Trivial.
By Theorem 9, we know that a well-filtered space
$X$
is locally compact iff
$X$
is core-compact. The following result shows that the condition that
$X$
is well-filtered can be weakened to the one that
$X$
is only
$\omega$
-well-filtered.
Theorem 28 (Xu et al. Reference Xu, Shen, Xi and Zhao2020a, Theorem 3.15). Let
$X$
be an
$\omega$
-well-filtered space. Then,
$X$
is locally compact iff
$X$
is core-compact.
Carrying out a proof similar to that of Proposition 7, we get the following result.
Proposition 29.
The
$\omega$
-well-filteredness is both closed-hereditary and saturated-hereditary.
Using (
$\omega$
-)Rudin sets and (
$\omega$
-)
$\mathsf{WD}$
sets, the following decomposition theorem for the sobriety was obtained in Xu et al., (Reference Xu, Shen, Xi and Zhao2020a,b).
Theorem 30 (Xu et al. Reference Xu, Shen, Xi and Zhao2020a, Theorem 5.11) (Xu et al. Reference Xu, Shen, Xi and Zhao2020b, Theorem 6.6). For a
$T_0$
-space
$X$
, the following conditions are equivalent:
-
(1)
$X$
is sober.
-
(2)
$X$
is a well-filtered and Rudin space.
-
(3)
$X$
is a well-filtered and WD space.
-
(4)
$X$
is an
$\omega$
-well-filtered and
$\omega$
-Rudin space.
-
(5)
$X$
is an
$\omega$
-well-filtered and
$\omega$
-WD space.
4. Simultaneous Convergence and
$\omega$
-Rudin Sets
Definition 31.
Let
$X$
be a
$T_0$
-space and
$A$
a nonempty subset of
$X$
. A net
$(x_e)_{e\in E}$
in
$X$
is said to converge simultaneously to all points of
$A$
, if
$(x_e)_{e\in E}$
converges to
$a$
for any
$a\in A$
.
Proposition 32.
Let
$X$
be a
$T_0$
-space
$X$
,
$A$
a nonempty subset of
$X$
and
$(x_e)_{e\in E}$
a net in
$X$
. Then, we have the following conclusions:
-
(1)
$(x_e)_{e\in E}$
simultaneously converges to all points of
$A$
iff for any
$U\in \mathscr O(X)$
,
$A\cap U\neq \emptyset$
implies there is
$e_U\in E$
such that
$x_e\in U$
for all
$e\geq e_U$
. -
(2)
$(x_e)_{e\in E}$
simultaneously converges to all points of
$A$
iff the net
$(\overline {\{x_e\}})_{e\in E}$
converges to
$\overline {A}$
in the Hoare power space
$P_H(X)$
. -
(3) If there is a net
$(s_d)_{d\in D}$
in
$\overline {A}$
such that
$(s_d)_{d\in D}$
simultaneously converges to all points of
$A$
, then
$A\in {\mathsf{Irr}} (X)$
and
$\overline {A}=\overline {\{s_d : d\geq d^{\prime }\}}$
for any
$d^{\prime }\in D$
. -
(4) If
$A\not \in {\mathsf{Irr}} (X)$
, then there is no net
$(s_d)_{d\in D}$
in
$\overline {A}$
such that
$(s_d)_{d\in D}$
simultaneously converges to all points of
$A$
. -
(5)
$(x_e)_{e\in E}$
simultaneously converges to all points of
$A$
iff
$(x_e)_{e\in E}$
simultaneously converges to all points of
$\overline {A}$
.
Proof.
(1): Assume that
$(x_e)_{e\in E}$
simultaneously converges to all points of
$A$
and
$U\in \mathscr O(X)$
. If
$A\cap U\neq \emptyset$
, then we can select a point
$a\in A\cap U$
. As
$(x_e)_{e\in E}$
simultaneously converges to all points of
$A$
,
$a$
is a convergence point of
$(x_e)_{e\in E}$
, and hence, there is
$e_U\in E$
such that
$x_e\in U$
for all
$e\geq e_U$
. Conversely, suppose that for any
$U\in \mathscr O(X)$
with
$U\cap A\neq \emptyset$
, there is
$e_U\in E$
such that
$x_e\in U$
for all
$e\geq e_U$
. Then, for any
$a\in A$
and
$V\in \mathscr O(X)$
with
$a\in V$
, we have
$A\cap V\neq \emptyset$
. Hence, there is
$e_V\in E$
such that
$x_e\in V$
for all
$e\geq e_V$
, proving that
$a$
is a convergence point of
$(x_e)_{e\in E}$
. Thus,
$(x_e)_{e\in E}$
simultaneously converges to all points of
$A$
.
(2): Assume that
$(x_e)_{e\in E}$
in
$X$
,
$(x_e)_{e\in E}$
simultaneously converges to all points of
$A$
. Let
$\bigcap \limits _{i=1}^{n}\Diamond U_n$
be a basic open set in
$P_H(X)$
with
$\overline {A}\in \bigcap \limits _{i=1}^{n}\Diamond U_n$
, where
$\{U_1, U_2, \ldots , U_n\}\subseteq \mathscr O(X)$
. Then, for each
$1\leq i\leq n$
,
$A\cap U_i\neq \emptyset$
and hence by (1) there is
$e_i\in E$
such that
$x_e\in U_i$
for all
$e\geq e_i$
. By the directness of
$E$
, there is
$e^*\in E$
with
$\{e_1, e_2, \ldots , e_n\}\subseteq \mathord {\downarrow }_E e^*$
. Then, for any
$e\geq e^*$
,
$x_e\in \bigcap \limits _{i=1}^{n}U_n$
and
$\overline {\{x_e\}}\in \Diamond \bigcap \limits _{i=1}^{n}U_n\subseteq \bigcap \limits _{i=1}^{n}\Diamond U_n$
, proving that
$(\overline {\{x_e\}})_{e\in E}$
converges to
$\overline {A}$
in
$P_H(X)$
. Conversely, if
$(\overline {\{x_e\}})_{e\in E}$
converges to
$\overline {A}$
in
$P_H(X)$
, then for any
$a\in A$
and
$U\in \mathscr O(X)$
with
$a\in U$
, we have
$\overline {A}\in \Diamond U\in \mathscr O(P_H(X))$
. Since
$(\overline {\{x_e\}})_{e\in E}$
converges to
$\overline {A}$
in
$P_H(X)$
, there is
$e^\prime \in E$
such that for all
$e\geq e^\prime$
$\overline {\{x_e\}}\in \Diamond U$
or, equivalently,
$x_e\in U$
. Hence,
$(x_e)_{e\in E}$
converges to
$a$
. Thus,
$(x_e)_{e\in E}$
simultaneously converges to all points of
$A$
.
(3): Suppose that
$U_i \in \mathscr O(X)$
and
$U_i\cap A\neq \emptyset$
(
$i=1, 2$
). Select a point
$a_i\in A\cap U_i$
(
$i=1, 2$
). Then
$a_1, a_2$
are two convergence points of
$(s_d)_{d\in D}$
, and hence, there is
$d_i\in D$
such that
$s_d\in U_i$
for all
$d\geq d_i$
(
$i=1, 2$
). By the directness of
$D$
, there is
$d_3\in D$
with
$d_1\leq d_3$
and
$d_2\leq d_3$
. Then,
$s_d\in U_1\cap U_2$
for any
$n\geq n_3$
. By
$s_{d_3}\in \overline {A}$
and
$s_{d_3}\in U_1\cap U_2$
, we get
$A\cap U_1\cap U_2\neq \emptyset$
. Hence,
$A\in {\mathsf{Irr}} (X)$
.
Clearly,
$\overline {\{s_d : d\geq d^{\prime }\}}\subseteq \overline {A}$
. On the other hand, for any
$s\in \overline {A}$
and
$U\in \mathscr O(X)$
with
$s\in U$
, we have
$A\cap U\neq \emptyset$
and hence by (1) there is
$d_U\in D$
such that
$s_d\in U$
for all
$d\geq d_U$
. By the directedness of
$D$
, there is
$d^{*}\in D$
such that
$d_U\leq d^{*}$
and
$d^{\prime }\leq d^{*}$
. Then,
$s_{d^*}\in \{s_d : d\geq d^{\prime }\}\cap U$
. Thus,
$s\in \overline {\{s_d : d\geq d^{\prime }\}}$
. So,
$\overline {A}\subseteq \overline {\{s_d : d\geq d^{\prime }\}}$
, and hence,
$\overline {A}=\overline {\{s_d : d\geq d^{\prime }\}}$
.
(4): By (3).
(5): By (2),
$(x_e)_{e\in E}$
simultaneously converges to all points of
$A$
iff the net
$(\overline {\{x_e\}})_{e\in E}$
converges to
$\overline {A}$
in
$P_H(X)$
iff
$(x_e)_{e\in E}$
simultaneously converges to all points of
$\overline {A}$
by
$\overline {\overline {A}}=\overline {A}$
and (2) again.
As
$X^s$
is a subspace of
$P_H(X)$
, by Proposition 32, we have the following.
Corollary 33.
Let
$X$
be a
$T_0$
-space
$X$
,
$A$
a nonempty closed subset of
$X$
and
$(x_e)_{e\in E}$
a net in
$X$
. Then, the following conditions are equivalent:
-
(1)
$(x_e)_{e\in E}$
simultaneously converges to all points of
$A$
. -
(2)
$A\in {\mathsf{Irr}}_c(X)$
and the net
$(\overline {\{x_e\}})_{e\in E}$
converges to
$\overline {A}$
in the Hoare power space
$P_H(X)$
. -
(3)
$A\in {\mathsf{Irr}}_c(X)$
and the net
$(\overline {\{x_e\}})_{e\in E}$
converges to
$A$
in the sobrification
$X^s$
.
For a
$T_0$
-space
$X$
and
$(x, U)\in X\times \mathscr O(X)$
, we have that
$x\in U$
iff
$\overline {\{x\}}\in \Diamond U$
iff
$\mathord {\uparrow } x\in \Box U$
. From this fact, we directly get the following.
Lemma 34.
Let
$X$
be a
$T_0$
-space
$X$
,
$x\in X$
and
$(x_e)_{e\in E}$
a net in
$X$
. Then, the following conditions are equivalent:
-
(1) The net
$(x_e)_{e\in E}$
converges to
$x$
in
$X$
. -
(2) The net
$(\overline {\{x_e\}})_{e\in E}$
converges to
$\overline {\{x\}}$
in the Hoare power space
$P_H(X)$
. -
(3) The net
$(\overline {\{x_e\}})_{e\in E}$
converges to
$\overline {\{x\}}$
in the sobrification
$X^s$
. -
(4) The net
$(\mathord {\uparrow } x_e)_{e\in E}$
converges to
$\mathord {\uparrow } x$
in the Smyth power space
$P_S(X)$
.
In Xu et al. (Reference Xu, Miao and Li2025), the following useful lemma was given.
Lemma 35 (Xu et al. Reference Xu, Miao and Li2025, Lemma 4.6). Suppose that
$X$
is a
$T_0$
-space and a sequence
$(K_n)_{n\in \mathbb{N}}$
converges to
$K$
in its Smyth power space
$P_S(X)$
. For any
$n\in \mathbb{N}$
, let
$G_n=K\cup \bigcup _{m\geq n}K_m$
. Then,
-
(1)
$G_{n+1} \subseteq G_n$
for any
$n\in \mathbb{N}$
. -
(2)
$G_n\in \mathsf{K}(X)$
for any
$n\in \mathbb{N}$
. -
(3)
$K=\bigcap _{n\in \mathbb{N}}G_n$
.
From Lemmas 34 and 35, we deduce the following.
Corollary 36.
Suppose that
$X$
is a
$T_0$
-space and
$(x_n)_{n\in \mathbb{N}}$
is a sequence converging to a point
$x$
in
$X$
. For any
$n\in \mathbb{N}$
, let
$H_n=\mathord {\uparrow } x\cup \bigcup _{m\geq n}\mathord {\uparrow } x_m$
. Then,
-
(1)
$H_{n+1} \subseteq H_n$
for any
$n\in \mathbb{N}$
. -
(2)
$H_n\in \mathsf{K}(X)$
for any
$n\in \mathbb{N}$
. -
(3)
$\mathord {\uparrow } x=\bigcap _{n\in \mathbb{N}}H_n$
.
Proposition 37.
Let
$X$
be a
$T_0$
-space,
$A$
an irreducible closed set of
$X$
and
$\{a_n : n\in \mathbb{N}\}\subseteq A$
. Suppose that the sequence
$(\overline {\{a_n\}})_{n\in \mathbb{N}}$
converges to
$A$
in the sobrification
$X^s$
. For any
$n\in \mathbb{N}$
, let
$K_n=\uparrow \{a_m : m\geq n\}$
. Then,
-
(1)
$K_{n+1} \subseteq K_n$
for any
$n\in \mathbb{N}$
. -
(2)
$K_n\in \mathsf{K}(X)$
for any
$n\in \mathbb{N}$
. -
(3)
$A\in m(\{K_n : n\in \mathbb{N}\})$
.
Therefore,
$A$
is an
$\omega$
-Rudin set of
$X$
.
Proof.
Obviously,
$K_{n+1} \subseteq K_n$
for any
$n\in \mathbb{N}$
. For any
$n\in \mathbb{N}$
, by Corollary 33, the sequence
$(a_l)_{l\in \mathbb{N}}$
converges to
$a_n$
in
$X$
. It follows from Corollary 36 that
$K_n=\mathord {\uparrow } a_n\cup \bigcup _{m\geq n}\mathord {\uparrow } a_m\in \mathsf{K}(X)$
.
Clearly,
$A\in M(\{K_n : n\in \mathbb{N}\})$
. For any
$B\in \Gamma (X)$
, if
$B$
is a proper closed subset of
$A$
, then there is
$a\in A\cap (X\setminus B)$
. As
$(a_n)_{n\in \mathbb{N}}$
is converges to
$a$
in
$X$
and
$a\in X\setminus B\in \mathscr O(X)$
, there is
$m_1\in \mathbb{N}$
such that
$a_n\in X\setminus B$
for all
$n\geq m_1$
. Then, for any
$n\geq m_1$
,
$K_n\cap B=\emptyset$
. So,
$B\notin M(\{K_n : n\in \mathbb{N}\})$
. Thus,
$A\in m(\{K_n : n\in \mathbb{N}\})$
. Therefore,
$A$
is an
$\omega$
-Rudin set of
$X$
.
Now, we give the following characterization of
$\omega$
-Rudin sets.
Theorem 38.
Let
$X$
be a
$T_0$
-space and
$A$
a nonempty closed set of
$X$
. Then, the following conditions are equivalent:
-
(1)
$A$
is an
$\omega$
-Rudin set of
$X$
. -
(2) There is
$\{a_n : n\in \mathbb{N}\}\subseteq A$
such that the sequence
$(a_n)_{n\in \mathbb{N}}$
simultaneously converges to all points of
$A$
. -
(3) There is
$\{a_n : n\in \mathbb{N}\}\subseteq A$
such that the sequence
$(\overline {\{a_n\}})_{n\in \mathbb{N}}$
converges to
$A$
in the Hoare power space
$P_H(X)$
. -
(4)
$A\in {\mathsf{Irr}}_c(X)$
and there is
$\{a_n : n\in \mathbb{N}\}\subseteq A$
such that the sequence
$(\overline {\{a_n\}})_{n\in \mathbb{N}}$
converges to
$A$
in the sobrification
$X^s$
.
Proof.
(1)
$\Rightarrow$
(2): By Proposition 27, there is a countable descending chain
$K_1\supseteq K_2\supseteq \ldots \supseteq K_n\supseteq \ldots$
of compact saturated subsets of
$X$
such that
$A\in m(\{K_n : n\in \mathbb{N}^{+}\}$
. For each
$n\in \mathbb{N}^{+}$
, choose
$a_n\in A\cap K_n$
. Now, we show that the sequence
$(a_n)_{n\in \mathbb{N}}$
simultaneously converges to all points of
$A$
. Assume, on the contrary, that there exists
$a\in A$
such that
$(a_n)_{n\in \mathbb{N}}$
does not converge to
$a$
. Then, there is
$U\in \mathscr O(X)$
such that
$a\in U$
and for any
$n\in \mathbb{N}^{+}$
, there is
$n^{\prime }\gt n$
with
$a_{n^{\prime }}\in X\setminus U$
, and hence
$a_{n^{\prime }}\in K_{n^{\prime }}\cap (X\setminus U)\cap A\subseteq K_n\cap (X\setminus U)\cap A$
. So,
$(X\setminus U)\cap A\in M(\{K_n : n\in \mathbb{N}^{+}\}$
. As
$A\in m(\{K_n : n\in \mathbb{N}^{+}\}$
, we have
$A=(X\setminus U)\cap A$
, being in contradiction with
$a\in A\cap U$
. Thus,
$(a_n)_{n\in \mathbb{N}}$
simultaneously converges to all points of
$A$
.
(2)
$\Leftrightarrow$
(3)
$\Leftrightarrow$
(4): By Proposition 32 and Corollary 33.
(3)
$\Rightarrow$
(1): By Proposition 37.
Remark 39. Let
$X$
be a
$T_0$
-space. Then,
$X$
is sober iff for any
$A\in {\mathsf{Irr}}_c(X)$
, there is a constant sequence
$(x)_{n\in \mathbb{N}}$
in
$A$
simultaneously converges to all points of
$A$
. Indeed, if
$X$
is sober and
$A\in {\mathsf{Irr}}_c(X)$
, then there is
$x\in X$
such that
$A=\overline {\{x\}}$
. It is easy to see that the constant sequence
$(x)_{n\in \mathbb{N}}$
in
$A$
simultaneously converges to all points of
$A$
. Conversely, suppose that for any
$A\in {\mathsf{Irr}}_c(X)$
, there is a constant sequence
$(x)_{n\in \mathbb{N}}$
in
$A$
simultaneously converges to all points of
$A$
. Then
$\overline {\{x\}}\subseteq A$
. For any
$a\in A$
, since
$(x)_{n\in \mathbb{N}}$
converges to
$a$
, we have that
$a\in \overline {\{x : n\in \mathbb{N}\}}=\overline {\{x\}}$
. So,
$A=\overline {\{x\}}$
. Thus,
$X$
is sober.
By Theorem 38 and Remark 39, the
$\omega$
-Rudin spaces can be considered as a kind of countable version of sober spaces.
From Theorem 38, we get the following characterization of
$\omega$
-Rudin spaces.
Theorem 40.
For a
$T_0$
-space
$X$
, the following conditions are equivalent:
-
(1)
$X$
is an
$\omega$
-Rudin space.
-
(2) For any
$A\in {\mathsf{Irr}}_c(X)$
, there is
$\{a_n : n\in \mathbb{N}\}\subseteq A$
such that the sequence
$(a_n)_{n\in \mathbb{N}}$
simultaneously converges to all points of
$A$
. -
(3) For any
$A\in {\mathsf{Irr}}_c(X)$
, there is
$\{a_n : n\in \mathbb{N}\}\subseteq A$
such that the sequence
$(\overline {\{a_n\}})_{n\in \mathbb{N}}$
converges to
$A$
in the Hoare power space
$P_H(X)$
. -
(4) For any
$A\in {\mathsf{Irr}}_c(X)$
, there is
$\{a_n : n\in \mathbb{N}\}\subseteq A$
such that the sequence
$(\overline {\{a_n\}})_{n\in \mathbb{N}}$
converges to
$A$
in the sobrification
$X^s$
.
By Proposition 32(3) and Theorem 38, we have the following.
Corollary 41.
Let
$X$
be a
$T_0$
-space. Then, we have the following conclusions:
-
(1) For any
$\omega$
-Rudin set
$A$
of
$X$
, there is a countable set
$\{a_n : n\in \mathbb{N}\}\subseteq A$
such that
$A=\overline {\{a_m : m\geq n\}}$
for any
$n\in \mathbb{N}$
. -
(2) If
$X$
is an
$\omega$
-Rudin space, then for any
$A\in {\mathsf{Irr}}_c(X)$
, there is a countable set
$\{a_n : n\in \mathbb{N}\}\subseteq A$
such that
$A=\overline {\{a_m : m\geq n\}}$
for any
$n\in \mathbb{N}$
.
Proposition 42.
For a poset
$P$
and
$A$
a nonempty lower set of
$P$
, the following conditions are equivalent:
-
(1)
$A$
is an
$\omega$
-Rudin set of
$(P, \alpha (P))$
. -
(2)
$A\in \operatorname{Id}(P)$
and
$A$
is countably lower-generated, that is, there is a countable set
$\{a_n : n\in \mathbb{N}\}$
such that
$A=\mathord {\downarrow } \{a_n : n\in \mathbb{N}\}$
. -
(3) There is a countable directed set
$D$
of
$P$
such that
$A=\mathord {\downarrow } D$
. -
(4) There is a countable chain
$C$
of
$P$
such that
$A=\mathord {\downarrow } C$
.
Proof.
(1)
$\Rightarrow$
(2): By Lemma 5,
$A\in \operatorname{Id}(P)$
. By Corollary 41, there is a countable set
$\{a_n : n\in \mathbb{N}\}\subseteq A$
such that
$A=\overline {\{a_n : n\in \mathbb{N}\}}=\mathord {\downarrow } \{a_n : n\in \mathbb{N}\}$
.
(2)
$\Rightarrow$
(3): Suppose that
$A\in \operatorname{Id}(P)$
and
$A=\mathord {\downarrow } \{a_n : n\in \mathbb{N}\}$
for some countable set
$\{a_n : n\in \mathbb{N}\}$
of
$P$
. Let
$D=\{a_n : n\in \mathbb{N}\}$
. Then,
$D$
is a countable directed set of
$P$
and
$A=\mathord {\downarrow } D$
.
(3)
$\Rightarrow$
(4): By Lemma 25.
(4)
$\Rightarrow$
(1): Assume that there is a countable chain
$C$
of
$P$
with
$A=\mathord {\downarrow } C$
. Then,
$A=\mathord {\downarrow } C=\textrm {cl}_{\alpha (P)} C\in \mathscr D_c^\omega ((P, \alpha (P))$
, and hence, by Lemma 22
$A$
is an
$\omega$
-Rudin set of
$(P, \alpha (P))$
.
By Lemma 5 and Proposition 42, we get the following.
Corollary 43.
For a poset
$P$
, the following conditions are equivalent:
-
(1)
$(P, \alpha (P))$
is an
$\omega$
-Rudin space.
-
(2) For any
$I\in \operatorname{Id}(P)$
, there is a countable directed set
$D$
of
$P$
such that
$I=\mathord {\downarrow } D$
. -
(3) For any
$I\in \operatorname{Id}(P)$
, there is a countable chain
$C$
of
$P$
such that
$I=\mathord {\downarrow } C$
.
Corollary 44.
For any countable poset
$P$
,
$(P, \alpha (P))$
is an
$\omega$
-Rudin space.
From Theorems 30 and 40 we deduce the following.
Theorem 45.
For a
$T_0$
-space
$X$
, the following conditions are equivalent:
-
(1)
$X$
is sober.
-
(2)
$X$
is well-filtered and for any
$A\in {\mathsf{Irr}}_c(X)$
, there is
$\{a_n : n\in \mathbb{N}\}\subseteq A$
such that the sequence
$(a_n)_{n\in \mathbb{N}}$
simultaneously converges to all points of
$A$
. -
(3)
$X$
is well-filtered and for any
$A\in {\mathsf{Irr}}_c(X)$
, there is
$\{a_n : n\in \mathbb{N}\}\subseteq A$
such that the sequence
$(\overline {\{a_n\}})_{n\in \mathbb{N}}$
converges to
$A$
in the Hoare power space
$P_H(X)$
. -
(4)
$X$
is well-filtered and for any
$A\in {\mathsf{Irr}}_c(X)$
, there is
$\{a_n : n\in \mathbb{N}\}\subseteq A$
such that the sequence
$(\overline {\{a_n\}})_{n\in \mathbb{N}}$
converges to
$A$
in the sobrification
$X^s$
. -
(5)
$X$
is
$\omega$
-well-filtered and for any
$A\in {\mathsf{Irr}}_c(X)$
, there is
$\{a_n : n\in \mathbb{N}\}\subseteq A$
such that the sequence
$(a_n)_{n\in \mathbb{N}}$
simultaneously converges to all points of
$A$
. -
(6)
$X$
is
$\omega$
-well-filtered and for any
$A\in {\mathsf{Irr}}_c(X)$
, there is
$\{a_n : n\in \mathbb{N}\}\subseteq A$
such that the sequence
$(\overline {\{a_n\}})_{n\in \mathbb{N}}$
converges to
$A$
in the Hoare power space
$P_H(X)$
. -
(7)
$X$
is
$\omega$
-well-filtered and for any
$A\in {\mathsf{Irr}}_c(X)$
, there is
$\{a_n : n\in \mathbb{N}\}\subseteq A$
such that the sequence
$(\overline {\{a_n\}})_{n\in \mathbb{N}}$
converges to
$A$
in the sobrification
$X^s$
.
By Proposition 10, Theorem 30, and Theorem 45, we get the following corollary.
Corollary 46.
For a complete lattice
$L$
, the following conditions are equivalent:
-
(1)
$\Sigma L$
is sober.
-
(2)
$\Sigma L$
is an
$\omega$
-Rudin space.
-
(3) For any nonempty Scott irreducible closed set
$A$
of
$L$
, there is
$\{a_n : n\in \mathbb{N}\}\subseteq A$
such that the sequence
$(a_n)_{n\in \mathbb{N}}$
simultaneously Scott converges to all points of
$A$
. -
(4) For any nonempty Scott irreducible closed set
$A$
of
$L$
, there is
$\{a_n : n\in \mathbb{N}\}\subseteq A$
such that the sequence
$(\overline {\{a_n\}})_{n\in \mathbb{N}}$
converges to
$A$
in the Hoare power space
$P_H(\Sigma L)$
. -
(5) For any nonempty Scott irreducible closed set
$A$
of
$L$
, there is
$\{a_n : n\in \mathbb{N}\}\subseteq A$
such that the sequence
$(\overline {\{a_n\}})_{n\in \mathbb{N}}$
converges to
$A$
in the sobrification
$(\Sigma L)^s$
.
In 2019, in a talkFootnote 1 given in National Institute of Education, Singapore, Achim Jung posed the following problem:
$\mathbf{Problem.}$
Is there a countable complete lattice that has a non-sober Scott topology?
Miao et al. (Reference Miao, Xi, Li and Zhao2023) constructed a countably infinite complete lattice whose Scott space is non-sober and thus gave a negative answer to the above problem.
Theorem 47 (Miao et al. Reference Miao, Xi, Li and Zhao2023, Corollary 3.6). There exists a countable complete lattice whose Scott space is non-sober.
The structure of such a countable complete lattice is somewhat complicated. Indeed, if
$L$
is a countable complete lattice whose Scott space is non-sober, then
$|L|=\omega$
. By Corollary 46, there exists an
$A\in {\mathsf{Irr}}_c(\Sigma \,\!L)$
such that for any subset
$\{a_n : n\in \mathbb{N}\}\subseteq A$
the sequence
$(\overline {\{a_n\}})_{n\in \mathbb{N}}$
does not converge to
$A$
in the sobrification
$(\Sigma \,\!L)^s$
, whence there is
$a\in A$
such that the sequence
$(a_n)_{n\in \mathbb{N}}$
does not converge to
$a$
in
$\Sigma \,\!L$
. It also provides an answer to the following question:
$\mathbf{Question.}$
Is there a
$T_0$
-space
$X$
in which there exists a countable subset
$C$
and
$x\in X$
such that
$x\in \overline {C}$
, but there is no sequence in
$C$
converging to
$x?$
In fact, let
$X=(\Sigma \,\!L)^s$
and
$\mathscr C=\{\overline {\{a\}} : a\in A\}$
. Then,
$\mathscr C$
is a countable subset of
$X$
and
$A\in \textrm {cl}_{X} \mathscr C$
(see Lemma 58 below) but there is no sequence
$(\overline {\{a_n\}})_{n\in \mathbb{N}}$
in
$\mathscr C$
converging to
$A$
in
$X$
. As
$A\in \textrm {cl}_{X} \mathscr C$
, there is a net
$(\overline {\{a_d\}})_{d\in D}$
in
$\mathscr C$
converging to
$A$
in
$X$
.
In the remainder of this section, we give some related examples. First, the space in the following example is an important well-filtered space, which is not an
$\omega$
-Rudin space.
Example 48.
Let
$X$
be an uncountably infinite set and
$X_{coc}$
the space equipped with the co-countable topology (the empty set and the complements of countable subsets of
$X$
are open). Then,
-
(a)
$\Gamma (X_{coc})=\{\emptyset , X\}\cup X^{(\leq \omega )}$
and
$X_{coc}$
is a
$T_1$
-space.
-
(b)
${\mathsf{Irr}}_c(X_{coc})=\{\{x\} : x\in X\}\cup \{X\}$
and
$\mathord {\mathsf{K}} (X_{coc})=X^{(\lt \omega )}\setminus \{\emptyset \}$
.It is easy to see that
${\mathsf{Irr}}_c(X_{coc})=\{\{x\} : x\in X\}\cup \{X\}$
. Clearly, every finite subset is compact. Conversely, if
$C\subseteq X$
is infinite, then
$C$
has an infinite countable subset
$\{c_n : n\in \mathbb{N}\}$
. Let
$C_0=\{c_n : n\in \mathbb{N}\}$
and
$U_m=(X\setminus C_0)\cup \{c_m\}$
for each
$m\in \mathbb{N}$
. Then,
$\{U_n : n\in \mathbb{N}\}$
is an open cover of
$C$
, but has no finite subcover. Hence,
$C$
is not compact. Thus,
$\mathord {\mathsf{K}} (X_{coc})=X^{(\lt \omega )}\setminus \{\emptyset \}$
. -
(c)
$X_{coc}$
is well-filtered.
Suppose that
$\{F_d : d\in D\}\subseteq \mathord {\mathsf{K}} (X_{coc})$
is a filtered family and
$U\in \mathscr O(X_{coc})$
with
$\bigcap _{d\in D}F_d\subseteq U$
. Then, by (b) all
$F_d$
are finite. As
$\{F_d : d\in D\}$
is filtered, it has a least element
$F_{d_0}$
, and hence
$F_{d_0}=\bigcap _{d\in D}F_d\subseteq U$
, proving that
$X_{coc}$
is well-filtered.
-
(d)
$X_{coc}$
is not an
$\omega$
-Rudin space and hence not a sober space. By (b)
$X\in {\mathsf{Irr}}_c(X_{coc})$
. As
$X$
is uncountable,
$X\neq \{a_n : n\in \mathbb{N}\}=\overline {\{a_n : n\in \mathbb{N}\}}$
for any countable subset
$\{a_n : n\in \mathbb{N}\}\subseteq X$
, and hence,
$X$
is not an
$\omega$
-Rudin set by Corollary 41
. Thus,
$X_{coc}$
is not an
$\omega$
-Rudin space. So, it is not sober by Theorem 30.
Now, we give two typical non-sober spaces which are
$\omega$
-Rudin spaces.
Example 49 (Johnstone space). Let
$\mathbb{J}=\mathbb{N}^{+}\times (\mathbb{N}^{+}\cup \{\infty \})$
with ordering defined by
$(j, k)\leq (m, n)$
iff
$j = m$
and
$k \leq n$
, or
$n =\infty$
and
$k\leq m$
(see Figure 1).
$\mathbb{J}$
is a well-known dcpo constructed by Johnstone (Reference Johnstone1981), which is the first given dcpo whose Scott space is non-sober. We will show that the Johnstone space is an
$\omega$
-Rudin space.
Johnstone’s dcpo
$\mathbb{J}$
.
It is easy to verify that
$\mathbb{J}$
is a dcpo, and hence, the Johnstone space
$\Sigma \,\!\mathbb{J}$
is a
$d$
-space. Clearly,
$\mathbb{J}_{max}=\{(n, \infty ) : n\in \mathbb{N}^{+} \}$
is the set of all maximal elements of
$\mathbb{J}$
.
The following three conclusions about
$\Sigma \,\!\mathbb{J}$
are known (see, e.g.,, (Lu et al. Reference Lu and Li2017, Example 3.1) and (Miao et al. Reference Miao, Li and Zhao2021, Lemma 3.1)):
-
(a)
${\mathsf{Irr}}_c (\Sigma \,\!\mathbb{J})=\{\overline {\{x\}}=\mathord {\downarrow }_{\mathbb{J}} x : x\in \mathbb{J}\}\cup \{\mathbb{J}\}$
. -
(b)
$\mathsf{K}(\Sigma \,\!\mathbb{J})=(2^{\mathbb{J}_{max}} \setminus \{\emptyset \})\bigcup \mathbf{Fin}\,\!\mathbb{J}$
. -
(c)
$\Sigma \,\!\mathbb{J}$
is not
$\omega$
-well-filtered and hence non-sober (cf. (Goubault, Reference Goubault-Larrecq2013, Exercise 8.3.9)). Let
$\mathscr G=\{\mathbb{J}\setminus F : F\in (\mathbb{J}_{max})^{(\lt \omega )}\}$
. Then, by (b),
$\mathscr G\subseteq \mathsf{K}(\Sigma \,\!\mathbb{J})$
is a countable filtered family and
$\bigcap \mathscr{G}=\bigcap _{F\in (\mathbb{J}_{max})^{(\lt \omega )}} (\mathbb{J}\setminus F)=\mathbb{J}_{max}\setminus \bigcup (\mathbb{J}_{max})^{(\lt \omega )}=\emptyset$
, but
$\mathbb{J}\setminus F=\emptyset$
for no
$F\in (\mathbb{J}_{max})^{(\lt \omega )}$
. So,
$\Sigma \,\!\mathbb{J}$
is not
$\omega$
-well-filtered, and hence, it is neither well-filtered nor sober by Theorem 30.
Now, we show that
$\Sigma \,\!\mathbb{J}$
is an
$\omega$
-Rudin space. As
${\mathsf{Irr}}_c (\Sigma \,\!\mathbb{J})=\{\overline {\{x\}} : x\in \mathbb{J}\}\cup \{\mathbb{J}\}$
, by Lemma 22
, we only need to prove that
$\mathbb{J}$
is an
$\omega$
-Rudin set. For each
$n\in \mathbb{N}^{+}$
, let
$a_n=(n, \infty )$
. Then, the sequence
$(a_n)_{n\in \mathbb{N}^{+}}$
simultaneously converges to all points of
$\mathbb{J}$
. In fact, for any nonempty
$U\in \sigma (\mathbb{J})$
, we have
$(m, \infty )\in U$
for some
$m\in \mathbb{N}^{+}$
. By
$(m, \infty )=\bigvee _{n\in \mathbb{N}^{+}}(m, n)\in U\in \sigma (\mathbb{J})$
, there is
$l\in \mathbb{N}^{+}$
such that
$\{(m, n) : n\geq l\}\subseteq U$
and hence
$a_n=(n, \infty )\in U$
for all
$n\geq l$
. By Proposition 32(1),
$(a_n)_{n\in \mathbb{N}^{+}}$
simultaneously converges to all points of
$\mathbb{J}$
. Thus,
$\mathbb{J}$
is an
$\omega$
-Rudin set by Theorem 38. Therefore,
$\Sigma \,\!\mathbb{J}$
is an
$\omega$
-Rudin space.
Example 50.
Let
$X$
be a countable infinite set and
$X_{cof}$
the space equipped with the co-finite topology (the empty set and the complements of finite subsets of
$X$
are open). Then,
-
(a)
$\Gamma (X_{cof})=\{\emptyset , X\}\cup X^{(\lt \omega )}$
and
${\mathsf{Irr}}_c(X_{cof})=\{\{x\} : x\in X\}\cup \{X\}$
. -
(b)
$X_{cof}$
is a
$T_1$
-space, and hence a
$d$
-space.
-
(c)
$X_{cof}$
is second-countable, and hence
$X^s$
,
$P_H(X)$
, and
$P_S(X)$
are second-countable by Lemma 13
. -
(d)
$X_{cof}$
is an
$\omega$
-Rudin space.
By (a) and Lemma 22 , we only need to show that
$X$
is an
$\omega$
-Rudin set. Let
$X=\{x_n : n\in \mathbb{N}\}$
. Then, for any
$x\in X$
and
$U=X\setminus F\in \mathscr O(X_{cof})$
with
$x\in U\setminus F$
, there is
$m\in \mathbb{N}$
such that
$F\cap \{x_n : n\geq m\}=\emptyset$
(note that
$F$
is finite), or equivalently,
$x_n\in U=X\setminus F$
for all
$n\geq m$
. Hence,
$(x_n)_{n\in \mathbb{N}}$
simultaneously converges to all points of
$X$
. Thus
$X_{cof}$
is an
$\omega$
-Rudin set by Theorem 38. So,
$X_{cof}$
is an
$\omega$
-Rudin space.
-
(e)
$\mathord {\mathsf{K}} (X_{cof})=2^X\setminus \{\emptyset \}$
and hence
$X_{cof}$
is both compact and locally compact.
-
(f)
$X_{cof}$
is not sober (in fact, not
$\omega$
-well-filtered). Let
$\mathscr{K}_{X}=\{X\setminus F : F\in X^{(\lt \omega )}\}$
. Then, by (e)
$\mathscr{K}_{X} \subseteq \mathord {\mathsf{K}} (X_{cof})$
is a countable filtered family and
$\bigcap \mathscr{K}_X=X\setminus \bigcup X^{(\lt \omega )}=X\setminus X=\emptyset$
, but
$X\setminus F=\emptyset$
for no
$ F\in X^{(\lt \omega )}$
. Thus,
$X_{cof}$
is not
$\omega$
-well-filtered, and consequently,
$X_{cof}$
is neither well-filtered nor sober.
The following example shows that a Rudin space may not be an
$\omega$
-Rudin space.
Example 51.
Let
$\mathbb{N}$
be the chain of natural numbers with the usual order and
$P=\mathbb{N}^\omega$
be the product of countable infinitely many copies of
$\mathbb{N}$
. Then,
-
(a)
$P$
is not a dcpo and hence
$(P, \alpha (P))$
is not a
$d$
-space.
Let
$D=\{(n, n, \ldots , n,\ldots ) : n\lt \omega \}$
. Then,
$D$
is a countable directed set of
$P$
and
$\vee D$
does not exist in
$P$
. Hence,
$P$
is not a dcpo. Thus,
$(P, \alpha (P))$
is not a
$d$
-space.
-
(b)
${\mathsf{Irr}}_c((P, \alpha (P)))=\operatorname{Id}(P)$
and
$\mathsf{K}((P, \alpha (P)))=\mathbf{Fin}\,\!P$
by Lemma 5
. -
(c)
$(P, \alpha (P))$
is a Rudin space by Proposition 18
. -
(d)
$(P, \alpha (P))$
is not an
$\omega$
-well-filtered space.
For each
$n\lt \omega$
, let
$K_n=\mathord {\uparrow } (n, n, \ldots , n, \ldots )$
. Then,
$\{K_n : n\lt \omega \}\subseteq \mathsf{K}((P, \alpha (P)))$
is a countable filtered family and
$\bigcap _{n\lt \omega }K_n=\emptyset$
, but
$K_n=\emptyset$
for no
$n\lt \omega$
. Hence,
$(P, \alpha (P))$
is not an
$\omega$
-well-filtered space.
-
(e)
$(P, \alpha (P))$
is not an
$\omega$
-Rudin space.
Clearly,
$P$
is directed and hence
$P\in {\mathsf{Irr}}_c((P, \alpha (P)))$
by (b). Now, we show that
$P$
is not an
$\omega$
-Rudin set. Suppose that
$\{\varphi _n : n\in \mathbb{N}\}$
is a countable set of
$P$
. For each
$n\lt \omega$
, select an
$l_n\in \mathbb{N}$
with
$\varphi _n(n)\lt l_n$
. Define
$\varphi ^*\in \mathbb{N}^\omega$
by
$\varphi ^*(n)=l_n$
for each
$n\lt \omega$
. Then,
$\varphi ^*\not \in \mathord {\downarrow } \{\varphi _n : n\in \mathbb{N}\}$
. So, there is no countable set
$C$
with
$P=\mathord {\downarrow } C$
. It follows from Proposition 42 that
$(P, \alpha (P))$
is not an
$\omega$
-Rudin space.
-
(f)
$(P, \alpha (P))$
is first-countable but not second-countable.
Obviously,
$(P, \alpha (P))$
is first-countable. Now, we show that
$(P, \alpha (P))$
is not second-countable. Suppose that
$\mathscr B\subseteq \alpha (P)$
is a base for
$(P, \alpha (P))$
. Then, for each
$x\in P$
, there is
$U_x\in \mathscr B$
such that
$x\in U_x\subseteq \mathord {\uparrow } x$
. Then,
$\mathord {\uparrow } x=U_x$
and hence
$\mathord {\uparrow } x\in \mathscr B$
. So,
$\{\mathord {\uparrow } x : x\in P\}\subseteq \mathscr B$
, and consequently,
$|\mathscr B|\geq |P|=2^{\omega }\gt \omega$
. Therefore,
$(P, \alpha (P))$
is not second-countable.
Finally, we give a
$T_0$
-space which satisfies condition (2) of Corollary 41 but is not an
$\omega$
-Rudin space.
Example 52.
Let
$\mathscr{L}=\mathbb{N}\times \mathbb{N}\times (\mathbb{N}\cup \{\infty \})$
. Define an order
$\leq$
on
$\mathscr L\,$
as follows:
$(i_1, j_1, k_1)\leq (i_2, j_2, k_2)$
if and only if:
(1)
$i_1=i_2, j_1=j_2, k_1\leq k_2$
; or
(2)
$i_1=i_2, j_1=j_2, k_2=\infty$
; or
(3)
$i_2=i_1+1, k_1\leq j_2, k_2=\infty$
.
$\mathscr L\,$
is a known dcpo given in (Jia Reference Jia2018, Example 2.6.1). It can be easily depicted as in Figure
2
taken from Jia (Reference Jia2018).
-
(a) It was shown in (Jia Reference Jia2018, Example 2.6.1) that
$\mathscr L\,$
is a dcpo and
$\Sigma \,\!\mathscr{L}$
is well-filtered but non-sober.
-
(b)
${\mathsf{Irr}}_c(\Sigma \,\!\mathscr{L})=\{\overline {\{x\}} : x\in \mathscr L\,\}\cup \{\mathscr L\,\}$
(see (Xu et al. Reference Xu, Wen and Xi2023, Lemma 80)).
-
(c) For any
$A\in {\mathsf{Irr}}_c(\Sigma \,\!\mathscr{L})$
, there is a countable set
$\{a_n : n\in \mathbb{N}\}\subseteq A$
such that
$A=\overline {\{a_n : n\geq m\}}$
for any
$m\in \mathbb{N}$
.By (b),
$A=\overline {\{x\}}$
for some
$x\in \mathscr L\,$
or
$A=\mathscr L\,$
. For the first case, let
$a_n=x$
for any
$n\in \mathbb{N}$
. Then,
$A=\overline {\{x\}}=\overline {\{a_n : n\geq m\}}$
for any
$m\in \mathbb{N}$
. Now, we assume
$A=\mathscr L\,$
. As
$\mathscr L\,$
is countable. The set
$\mathscr L\,_{\operatorname{max}}=\{(i, j, \infty ) : (i, j)\in \mathbb{N}\times \mathbb{N}\}$
(the set of all maximal elements of
$\mathscr L\,$
) is countable. Let
$\mathscr L\,_{\operatorname{max}}=\{a_n : n\in \mathbb{N}\}$
. Then, it is easy to see that
$A=\mathscr L\,=\overline {\{a_n : n\geq m\}}$
for any
$m\in \mathbb{N}$
. -
(d)
$\Sigma \,\!\mathscr{L}$
is not a Rudin space and hence not an
$\omega$
-Rudin space.
By (a) and Theorem 30 ,
$\Sigma \,\!\mathscr{L}$
is not a Rudin space and hence not an
$\omega$
-Rudin space. Indeed, by the well-filteredness of
$\Sigma \mathscr{L}$
, one can directly get that the irreducible closed set
$\mathscr L\,$
is not a Rudin set of
$\Sigma \,\!\mathscr{L}$
. Assume, on the contrary, that
$\mathscr L\,$
is a Rudin set in
$\Sigma \,\!\mathscr{L}$
. Then, there is a filtered family
$\mathscr K\subseteq \mathord {\mathsf{K}}(\Sigma \,\!\mathscr{L})$
such that
$\mathscr L\,\in m(\mathscr K)$
. As
$\Sigma \,\!\mathscr{L}$
is well-filtered,
$\bigcap \mathscr K\cap \mathscr L\,\neq \emptyset$
. Select
$x\in \bigcap \mathscr K\cap \mathscr L\,$
. Then,
$\overline {\{x\}}=\mathord {\downarrow } x\in M(\mathscr K)$
. By the minimality of
$\mathscr L\,$
, we have
$\mathscr L\,=\mathord {\downarrow } x$
, a contradiction. So,
$\mathscr L\,$
is not a Rudin set.
-
(e)
$\Sigma \,\!\mathscr{L}$
is not first-countable by (a) and (Xu et al. Reference Xu, Shen, Xi and Zhao2020a, Theorem 4.2) (see Theorem 62 below).
The countable dcpo
$\mathscr L\,$
.
5. Some Basic Properties of
$\omega$
-Rudin Spaces
In this section, we discuss some basic properties of
$\omega$
-Rudin spaces. It will be shown that the property of being an
$\omega$
-Rudin space is retractive, productive, and closed-hereditary but is not saturated-hereditary.
Proposition 53.
The property of being an
$\omega$
-Rudin space is retractive.
Proof.
Suppose that
$Y$
is a retract of an
$\omega$
-Rudin space
$X$
. Then, there are continuous mappings
$f : X\longrightarrow Y$
and
$g : Y\longrightarrow X$
such that
$f\circ g=id_Y$
. Let
$B\in {\mathsf{Irr}}_c(Y)$
. Then, by Lemma 3,
$\overline {g(B)}\in {\mathsf{Irr}}_c(X)$
. As
$X$
is an
$\omega$
-Rudin space, by Theorem 40, there is
$\{a_n : n\in \mathbb{N}\}\subseteq \overline {g(B)}$
such that the sequence
$(a_n)_{n\in \mathbb{N}}$
simultaneously converges to all points of
$\overline {g(B)}$
. For each
$n\in \mathbb{N}$
, let
$b_n=f(a_n)$
. By the continuity of
$f$
, we have
$b_n=f(a_n)\in f(\overline {g(B)})\subseteq \overline {f(\overline {g(B)})}=\overline {f(g(B))}=\overline {B}=B$
for all
$n\in \mathbb{N}$
. Now, we show that
$(b_n)_{n\in \mathbb{N}}$
simultaneously converges to all points of
$B$
. For any
$b\in B$
and
$V\in \mathscr O(Y)$
with
$b\in V$
, we have
$g(b)\in g(B)\subseteq \overline {g(B)}$
and
$g(b)\in f^{-1}(V)\in \mathscr O(X)$
. As
$g(b)$
is a convergence point of
$\{a_n : n\in \mathbb{N}\}$
, there is
$n^*\in \mathbb{N}$
such that
$a_n\in f^{-1}(V)$
for all
$n\geq n^*$
and hence
$\{b_n=f(a_n) : n\geq n^*\}\subseteq f(f^{-1}(V))\subseteq V$
, proving that
$b$
is a convergence point of
$(b_n)_{n\in \mathbb{N}}$
. So,
$(b_n)_{n\in \mathbb{N}}$
simultaneously converges to all points of
$B$
. Thus,
$Y$
is
$\omega$
-Rudin by Theorem 40.
Theorem 54.
Let
$\{X_i:i\in I\}$
be a family of
$T_0$
-spaces. Then, the following two conditions are equivalent:
-
(1) The product space
$\prod _{i\in I}X_i$
is an
$\omega$
-Rudin space.
-
(2) For each
$i \in I$
,
$X_i$
is an
$\omega$
-Rudin space.
Proof.
For the convenience of proof, let
$X=\prod _{i\in I}X_i$
and
$p_i : X \rightarrow X_i$
be the
$i$
th projection (
$i\in I$
).
(1)
$\Rightarrow$
(2): For each
$i\in I$
, by Lemma 14,
$X_i$
is a retract of
$X$
and hence is an
$\omega$
-Rudin space by Proposition 53.
(2)
$\Rightarrow$
(1): Suppose
$A\in {\mathsf{Irr}}_c (X)$
. Then, by Lemma 4,
$A=\prod _{i\in I}p_i (A)$
and
$p_j(A)\in {\mathsf{Irr}}_c(X_j)$
for all
$j \in I$
. Hence, for each
$i\in I$
, by Theorem 40, there is
$\{a_n^i : n\in \mathbb{N}\}\subseteq p_i(A)$
such that the sequence
$(a_n^i)_{n\in \mathbb{N}}$
simultaneously converges to all points of
$p_i (A)$
. For each
$n\in \mathbb{N}$
, let
$a_n=(a_n^i)_{i\in I}$
. Then
$a_n\in \prod _{i\in I}p_i (A)=A$
for all
$n\in \mathbb{N}$
. Now, we show that the sequence
$(a_n)_{n\in \mathbb{N}}$
simultaneously converges to all points of
$A$
. Let
$a\in A$
and
$U\in \mathscr O(X)$
with
$a\in U$
. Then, there exists
$\{i_1, i_2, \ldots , i_m\}\subseteq I$
and
$(U_1, U_2, \ldots , U_m)\in \mathscr O(X_{i_1})\times \mathscr O(X_{i_2})\times \ldots \times \mathscr O(X_{i_m})$
such that
$a\in p_{i_1}^{-1}(U_1)\cap p_{i_2}^{-1}(U_2)\cap \ldots \cap p_{i_m}^{-1}(U_m)\subseteq U$
. For each
$1\leq k\leq m$
, we have
$p_{i_k}(a)\in p_{i_k} (A)$
and
$p_{i_k}(a)\in U_k$
, whence there is
$n_k\in \mathbb{N}$
such that
$a_n^{i_k}\in U_k$
for all
$n\geq n_k$
. Let
$n^\prime =\mbox{max}\{n_1, n_2, \ldots , n_m\}$
. Then,
$a_n=(a_n^i)_{i\in I}\in p_{i_1}^{-1}(U_1)\cap p_{i_2}^{-1}(U_2)\cap \ldots \cap p_{i_m}^{-1}(U_m)\subseteq U$
for all
$n\geq n^\prime$
, proving that
$a$
is a convergence point of
$(a_n)_{n\in \mathbb{N}}$
. So, the sequence
$(a_n)_{n\in \mathbb{N}}$
simultaneously converges to all points of
$A$
. Thus,
$X$
is an
$\omega$
-Rudin space by Theorem 40.
From Lemma 2 and Theorem 38, we directly deduce the following.
Proposition 55.
Let
$X$
be an
$\omega$
-Rudin space and
$A$
a nonempty closed subspace of
$X$
. Then,
$A$
is an
$\omega$
-Rudin space. Hence, the property of being an
$\omega$
-Rudin space is closed-hereditary.
It is well-known that the sobriety is saturated-hereditary (see, e.g., (Gierz et al. Reference Gierz, Hofmann, Keimel, Lawson, Mislove and Scott2003, Exercise O-5.16(ii))). From Propositions 7 and 29 we know that both the well-filteredness and the
$\omega$
-well-filteredness are saturated-hereditary. Combining
$X_{coc}$
(
$|X|\gt \omega$
) and
$\mathbb{N}_{cof}$
, we will give an example to show that unlike the sobriety and the (
$\omega$
-)well-filteredness, the property of being an
$\omega$
-Rudin space is not saturated-hereditary.
Example 56.
Suppose that
$X$
is a uncountable set and
$X\cap \mathbb{N}=\emptyset$
. Let
$Y=X\cup \mathbb{N}$
. Endow
$Y$
with the topology
$\{\emptyset , Y\}\bigcup \{U\cup V: U\in \mathscr O(X_{coc})\setminus \{\emptyset \}, V\in \mathscr O({\mathbb{N}_{cof}})\setminus \{\emptyset \}\}$
(note that
$U\cap V=\emptyset$
for any
$(U, V)\in \mathscr O(X_{coc})\times \mathscr O({\mathbb{N}_{cof}})$
). Then,
-
(a)
$\Gamma (Y)=\{\emptyset , Y\}\bigcup \{C\cup F : C\in X^{(\leq \omega )}, F\in \mathbb{N}^{(\lt \omega )}\}$
and hence
$Y$
is a
$T_1$
-space.
-
(b) For any nonempty open set
$W$
of
$Y$
,
$\mathbb{N}\setminus W$
is finite.
Since
$W\in \mathscr O(Y)\setminus \{\emptyset \}$
,
$W=(X\setminus C)\cup (\mathbb{N}\setminus F)$
for some
$(C, F)\in X^{(\leq \omega )}\times \mathbb{N}^{(\lt \omega )}$
(when
$C=\emptyset$
and
$F=\emptyset$
, we have
$W=Y$
). Then,
$\mathbb{N}\setminus W\subseteq \mathbb{N}\setminus (\mathbb{N}\setminus F)=F$
and hence
$\mathbb{N}\setminus W$
is finite.
-
(c)
${\mathsf{Irr}}_c(Y)=\{\{y\} : y\in Y\}\cup \{Y\}$
and
$Y$
is non-sober.
-
(d)
$\mathord {\mathsf{K}} (Y)=\mathord {\mathsf{K}} (X_{coc})\bigcup \mathord {\mathsf{K}} (\mathbb{N}_{cof})\bigcup \{K_1\cup K_2: K_1\in \mathord {\mathsf{K}} (X_{coc}), K_2\in \mathord {\mathsf{K}} (\mathbb{N}_{cof})\}=\{F\cup B : F\in X^{(\lt \omega )}\cup \{\emptyset \}, B\in 2^{\mathbb{N}}\}\setminus \{\emptyset \}$
.Assume that
$(F, B)\in (X^{(\lt \omega )}\cup \{\emptyset \})\times 2^{\mathbb{N}}$
and
$F\cup B\neq \emptyset$
. We show that
$A\cup B\in \mathord {\mathsf{K}} (Y)$
. Suppose that
$\{U_i\cup V_i: i\in I\}$
is an open cover of
$F\cup B$
in
$Y$
, where
$\{U_i : i\in I\}\subseteq \mathscr O(X_{coc})\setminus \{\emptyset \}$
and
$\{V_i : i\in I\}\subseteq \mathscr O({\mathbb{N}_{cof}})\setminus \{\emptyset \}$
. As
$X\cap \mathbb{N}=\emptyset$
, we have that
$F\subseteq \bigcup _{i\in I} U_i$
and
$B\subseteq \bigcup _{i\in I}V_i$
. Then, there is
$I_1\in I^{(\lt \omega )}$
such that
$F\subseteq \bigcup _{i\in I_1} U_i$
. By Example
50
(e), there is
$I_2\in I^{(\lt \omega )}$
such that
$B\subseteq \bigcup _{i\in I_2} V_i$
. Then,
$F\cup B\subseteq \bigcup _{i\in I_1\cup I_2}U_i\cup V_i$
. So
$F\cup B\in \mathord {\mathsf{K}} (Y)$
. Conversely, assume
$K\in \mathord {\mathsf{K}}(Y)$
. Let
$G=K\cap X$
and
$E=K\cap \mathbb{N}$
. Then
$K=K\cap Y=K\cap (X\cup \mathbb{N})=G\cup E$
. Now, we show that
$G\in \mathord {\mathsf{K}} (X_{coc})\cup \{\emptyset \}$
. Suppose that
$G\neq \emptyset$
and
$\{W_j : j\in J\}$
(where
$W_j\neq \emptyset$
for any
$j\in J$
) is an open cover of
$G$
in
$X_{coc}$
. For each
$j\in J$
, let
$\widetilde {W}_j=W_j\cup \mathbb{N}$
. Then,
$\{\widetilde {W}_j : j\in J\}$
is an open cover of
$K$
in
$Y$
, and hence, there is
$J_0\in J^{(\lt \omega )}$
such that
$K\subseteq \bigcup _{j\in J_0}\widetilde {W}_j$
, and consequently,
$G\subseteq \bigcup _{j\in J_0}W_j$
. Thus,
$G\in \mathord {\mathsf{K}} (X_{coc})$
, or equivalently,
$G\in X^{(\lt \omega )}\setminus \{\emptyset \}$
. Similarly, we can show that
$E=K\cap \mathbb{N}\in \mathord {\mathsf{K}} (\mathbb{N}_{cof})\cup \{\emptyset \}=2^{\mathbb{N}}$
. Hence,
$K=G\cup E\in \{F\cup B : F\in X^{(\lt \omega )}\cup \{\emptyset \}, B\in 2^{\mathbb{N}}\}\setminus \{\emptyset \}$
. -
(e)
$Y$
is not
$\omega$
-well-filtered.
By Example 50 (f),
$\mathbb{N}_{cof}$
is not
$\omega$
-well-filtered. Clearly,
$\mathbb{N}$
as a saturated subspace of
$Y$
is
$\mathbb{N}_{cof}$
(note that
$Y$
is
$T_1$
and hence the specialization order of
$Y$
is the discrete order). By Proposition 29
,
$Y$
is not
$\omega$
-well-filtered.
-
(f)
$Y$
is an
$\omega$
-Rudin space.
By (c), we only need to prove that there is
$\{y_n : n\in \mathbb{N}\}\subseteq Y$
such that the sequence
$(y_n)_{n\in \mathbb{N}}$
simultaneously converges to all points of
$Y$
. In fact, by (b), we know that the sequence
$(n)_{n\in \mathbb{N}}$
simultaneously converges to all points of
$Y$
. So,
$Y$
is an
$\omega$
-Rudin space.
-
(g) For any
$W\in \mathscr O(Y)$
,
$W$
as an open subspace of
$Y$
is an
$\omega$
-Rudin space.
We can suppose that
$W\neq \emptyset$
. If
$W=Y$
, then by (f)
$W$
is an
$\omega$
-Rudin space. If
$W\neq Y$
, then
$W=U\cup V$
for some
$U\in \mathscr O(X_{coc})\setminus \{\emptyset \}$
and
$V=\mathbb{N}\setminus F\in \mathscr O({\mathbb{N}_{cof}})\setminus \{\emptyset \}$
, where
$F$
is a finite subset of
$\mathbb{N}$
. Suppose that
$B\in {\mathsf{Irr}}_c(W)$
. Then,
$B=W\cap \textrm {cl}_Y B$
and
$\textrm {cl}_Y B\in {\mathsf{Irr}}_c(Y)$
by Lemma 2
. By (c)
$\textrm {cl}_Y B=Y$
or
$\textrm {cl}_Y B=\{y\}$
for some
$y\in Y$
. If
$\textrm {cl}_Y B=Y$
, then
$B=W\cap Y=W$
. Let
$\mathbb{N}\setminus F=\{a_n : n\lt \omega \}$
. Then
$\{a_n : n\lt \omega \}\subseteq B$
and by (b) the the sequence
$(a_n)_{n\in \mathbb{N}}$
simultaneously converges to all points of
$W$
. If
$\textrm {cl}_Y B=\{y\}$
for some
$y\in Y$
, then
$B=W\cap \{y\}=\{y\}$
(note that
$B\neq \emptyset$
), and hence, the constant sequence
$(y)_{n\in \mathbb{N}}$
simultaneously converges to all points of
$B$
. Thus,
$W$
is an
$\omega$
-Rudin space.
-
(h)
$X_{coc}$
as a saturated subspace of
$Y$
is not an
$\omega$
-Rudin space by Example
48
.
6. Fréchet Spaces and
$\omega$
-Rudin Spaces
A topological space
$X$
is called a Fréchet space if for every
$A\subseteq X$
and every
$x\in \overline {A}$
, there exists a sequence
$x_1, x_2, \ldots , x_n, \ldots$
of
$A$
converging to
$x$
.
The following result is well-known (see, e.g., (Engelking Reference Engelking1989, Theorem 1.6.14 and Exercise 2.1.H(b))).
Proposition 57. Every first-countable space is a Fréche space and every subspace of a Fréchet space is a Fréchet space.
As shown in (Engelking Reference Engelking1989, Example 1.6.18), a Fréche space may not be first-countable.
Lemma 58.
Let
$X$
be a
$T_0$
-space and
$A$
a nonempty subset of
$X$
. Then, the following three conditions are equivalent:
-
(1)
$\overline {A}\in \textrm {cl}_{P_H(X)}\{\overline {\{a\}} : a\in A\}$
. -
(2)
$A\in {\mathsf{Irr}} (X)$
and
$\overline {A}\in \textrm {cl}_{X^s}\{\overline {\{a\}} : a\in A\}$
. -
(3)
$A\in {\mathsf{Irr}} (X)$
.
Proof.
(1)
$\Rightarrow$
(2): As
$X^s$
is a subspace of
$P_H(X)$
, we only need to show that
$A\in {\mathsf{Irr}} (X)$
. Suppose that
$U, V\in \mathscr O(X)$
and
$A\cap U\neq \emptyset \neq A\cap V$
. Then,
$\overline {A}\in \Diamond U\cap \Diamond V\in \mathscr O(P_H(X))$
. By
$\overline {A}\in \textrm {cl}_{P_H(X)}\{\overline {\{a\}} : a\in A\}$
, there is
$a\in A$
with
$\overline {\{a\}}\in \Diamond U\cap \Diamond V$
, and hence,
$a\in A\cap U\cap V$
. So,
$A\in {\mathsf{Irr}} (X)$
.
(2)
$\Rightarrow$
(3): Trivial.
(3)
$\Rightarrow$
(1): Let
$\mathscr U\in \mathscr O(P_H(X))$
with
$\overline {A}\in \mathscr U$
. Then, there are finitely open sets
$U_1, U_2, \ldots , U_n$
of
$X$
such that
$\overline {A}\in \Diamond U_1\cap \Diamond U_2\cap \ldots \cap \Diamond U_n\subseteq \mathscr U$
, and hence,
$A\cap U_i\neq \emptyset$
(
$i=1, 2, \ldots , n$
). By the irreducibility of
$A$
, we have
$A\cap U_1\cap U_2\cap \ldots \cap U_n\neq \emptyset$
. Select any
$a\in A\cap U_1\cap U_2\cap \ldots \cap U_n$
. Then
$\overline {\{a\}}\in \Diamond U_1\cap \Diamond U_2\cap \ldots \cap \Diamond U_n\subseteq \mathscr U$
. Thus
$\overline {A}\in \textrm {cl}_{P_H(X)}\{\overline {\{a\}} : a\in A\}$
.
By Theorem 30, Theorem 40, and Lemma 58, we get the following.
Proposition 59 (Xu et al. Reference Xu, Miao and Li2025, Theorems 6.1 and 6.2). Let
$X$
be a
$T_0$
-space. Then, we have the following two conclusions:
-
(1) If its sobrification
$X^s$
(especially, its Hoare power space
$P_H(X)$
) is a Fréchet space, then
$X$
is an
$\omega$
-Rudin spaces.
-
(2) If
$X$
is an
$\omega$
-well-filtered space for which
$X^s$
(especially,
$P_H(X)$
) is a Fréchet space, then
$X$
is sober.
By Lemma 13, Proposition 57, and Proposition 59, we get the following two corollaries.
Corollary 60 (Xu et al. Reference Xu and Zhao2021, Theorems 5.6 and 5.9). Let
$X$
be a
$T_0$
-space. Then, we have the following two conclusions:
-
(1) If its sobrification
$X^s$
(especially, its Hoare power space
$P_H(X)$
) is first-countable, then
$X$
is an
$\omega$
-Rudin space.
-
(2) If
$X$
is an
$\omega$
-well-filtered space for which
$X^s$
(especially,
$P_H(X)$
) is first-countable, then
$X$
is sober.
Corollary 61 (Xu et al. Reference Xu and Zhao2021, Corollaries 5.7 and 5.11). (1) Every second-countable
$T_0$
-space is an
$\omega$
-Rudin space.
(2) Every second-countable
$\omega$
-well-filtered space is sober.
When
$X$
is only first-countable, the following result was given in Xu et al. (Reference Xu, Shen, Xi and Zhao2020a).
Theorem 62 (Xu et al. Reference Xu, Shen, Xi and Zhao2020a, Theorem 4.2). Let
$X$
be a first-countable
$T_0$
-space. Then, the following three conditions are equivalent:
-
(1)
$X$
is a sober space.
-
(2)
$X$
is a well-filtered space.
-
(3)
$X$
is an
$\omega$
-well-filtered
$d$
-space.
Corollary 63.
Suppose that
$P$
is a dcpo and
$\Sigma \,\!P$
is first-countable. Then, the following three conditions are equivalent:
-
(1)
$\Sigma \,\!P$
is sober.
-
(2)
$\Sigma \,\!P$
is well-filtered.
-
(3)
$\Sigma \,\!P$
is
$\omega$
-well-filtered.
By Proposition 59, Corollary 60, Corollary 61, and Theorem 62, we get the following.
Corollary 64.
Let
$X$
a
$T_0$
-space. Then, we have the following conclusions:
-
(1) If
$X$
is an
$\omega$
-well-filtered
$d$
-space but non-sober, then
$X$
is not first-countable.
-
(2) If
$X$
is well-filtered but non-sober, then
$X$
is not first-countable.
-
(3) If
$X$
is an
$\omega$
-well-filtered space but non-sober, then neither
$X^s$
nor
$P_H(X)$
is a Fréchet space.
-
(4) If
$X$
is an
$\omega$
-well-filtered space but non-sober, then neither
$X^s$
nor
$P_H(X)$
is first-countable.
-
(5) If
$X$
is an
$\omega$
-well-filtered space but non-sober, then
$X$
is not second-countable.
A Fréchet well-filtered space may not be sober as shown in the following example.
Example 65.
Let
$\mathscr{L}=\mathbb{N}\times \mathbb{N}\times (\mathbb{N}\cup \{\infty \})$
be the countable dcpo in Example
52
and let
$\mathscr L\,_{\operatorname{max}}=\{(i, j, \infty ) : (i, j)\in \mathbb{N}\times \mathbb{N}\}$
(the set of all maximal elements of
$\mathscr L\,$
). The following conclusions were proved in (Xu et al. Reference Xu, Miao and Li2025, Example 5.9):
-
(a)
$\mathscr L\,_{\operatorname{max}}$
, as a saturated subspace of
$\Sigma \,\!\mathscr L\,$
, is well-filtered (note that
$\Sigma \,\!\mathscr L\,$
is well-filtered but non-sober).
-
(b)
$\mathscr L\,_{\operatorname{max}}$
is a Fréchet space.
-
(c)
$\mathscr L\,_{\operatorname{max}}$
is not sober.
The example below (see also Example 51) shows that a first-countable
$T_0$
-space may not be an
$\omega$
-Rudin space (comparing it with Corollary 60(1) and Corollary 61(1)). It also shows that a first-countable
$\omega$
-well-filtered space may not be sober (comparing it with Corollary 60(2), Corollary 61(2) and Theorem 62).
Example 66.
Let
$P=[0, \omega _1)$
(recall that
$\omega _1$
is the first uncountable ordinal). Then,
-
(a)
$P$
is not a dcpo and hence
$\Sigma P$
is not a
$d$
-space.
Since
$P$
is directed and
$\vee P$
does not exist,
$P$
is not a dcpo. So
$\Sigma P$
is not a
$d$
-space, and hence it is neither well-filtered nor sober.
-
(b)
$\mathsf{K}(\Sigma P)=\{\mathord {\uparrow } x : x\in P\}$
. So
$\Sigma P$
is compact and locally compact.
For
$K\in \mathsf{K}(\Sigma P)$
, we have
$\operatorname{inf}\,K\in K$
, and hence
$K=\mathord {\uparrow } \operatorname{inf}\,K$
. Therefore,
$\mathsf{K}(\Sigma P)=\{\mathord {\uparrow } x : x\in P\}$
, and consequently,
$\Sigma P$
is compact and locally compact.
-
(c)
$\Sigma P$
is first-countable and hence a Fréchet space.
For
$\alpha \in P$
, if
$\alpha$
is a compact element of
$P$
, then
$\mathord {\uparrow } \alpha \in \sigma (P)$
and
$\{\mathord {\uparrow } \alpha \}$
is a base at
$\alpha$
. If
$\alpha$
is not a compact element of
$P$
, then
$[0, \alpha )$
is countable and
$\vee [0, \alpha )=\alpha$
. In this case,
$\{(\beta , \omega _1) : \beta \in [0, \alpha )\}$
is a countable base at
$\alpha$
. Thus
$\Sigma P$
is first-countable.
-
(d)
$\Gamma (\Sigma P)=\{\mathord {\downarrow } t : t\in P\}\cup \{\emptyset , P\}$
and
${\mathsf{Irr}}_c(\Sigma P)=\{\mathord {\downarrow } t : t\in P\}\cup \{P\}=\Gamma (\Sigma P)\setminus \{\emptyset \}$
.Assume
$A\in \Gamma (\Sigma P)\setminus \{\emptyset , P\}$
. Then, there is
$\beta \in P\setminus A$
, and hence,
$\beta$
is an upper bound of
$A$
in
$P$
. So
$\vee A$
exists in
$P$
. Since
$A$
is Scott closed and
$A$
is a chain, we have
$\vee A\in A$
, and consequently,
$A=\downarrow \vee A$
. So
$\Gamma (\Sigma P)=\{\mathord {\downarrow } t : t\in P\}\cup \{\emptyset , P\}$
. Clearly,
$P=[0, \omega _1)$
is irreducible in
$\Sigma P$
, and hence
${\mathsf{Irr}}_c(\Sigma P)=\{\mathord {\downarrow } t : t\in P\}\cup \{P\}=\Gamma (\Sigma P)\setminus \{\emptyset \}$
. -
(e)
$\Sigma P$
is
$\omega$
-well-filtered.
If
$\{\mathord {\uparrow } x_n : n\in \mathbb{N}\}\subseteq \mathsf{K}(\Sigma P)$
is countable filtered family and
$U\in \sigma (P)$
with
$\bigcap _{n\in \mathbb{N}}\mathord {\uparrow } x_n\subseteq U$
, then
$\{x_n : i\in \mathbb{N}\}$
is a countable subset of
$P=[0, \omega _1)$
. Since sup of a countable family of countable ordinal numbers is still a countable ordinal number, we have
$\beta =\operatorname{sup}\{x_n : n\in \mathbb{N}\}\in P$
, and hence
$\mathord {\uparrow } \beta =\bigcap _{n\in \mathbb{N}}\mathord {\uparrow } x_n\subseteq U$
. Therefore,
$\beta \in U$
, and consequently,
$x_n\in U$
for some
$n\in \mathbb{N}$
or, equivalently,
$\mathord {\uparrow } x_n\subseteq U$
, proving that
$\Sigma P$
is
$\omega$
-well-filtered.
-
(f)
$\Sigma P$
is a Rudin space.
By (d) and Lemma 17 , we only need to show that
$P$
is a Rudin set. As
$P$
is a chain,
$\{\mathord {\uparrow } x : x\in P\}\subseteq \mathsf{K}(\Sigma P)$
is a filtered family. Clearly,
$P\in M(\{\mathord {\uparrow } x : x\in P\})$
. If
$B$
is a nonempty Scott closed subset and
$B\neq P$
, then there is
$\alpha \in P$
such that
$\alpha \not \in B$
. So
$B\cap \mathord {\uparrow } \alpha =\emptyset$
, and consequently,
$B\not \in M(\{\mathord {\uparrow } x : x\in P\})$
. Thus,
$P\in m( \{\mathord {\uparrow } x : x\in P\}$
, proving that
$P$
is a Rudin set. Therefore,
$\Sigma P$
is a Rudin space.
-
(g)
$\Sigma P$
is not an
$\omega$
-Rudin space.
By (d)
$P\in {\mathsf{Irr}}_c(\Sigma P)$
. We show that
$P$
is not an
$\omega$
-Rudin set in
$\Sigma P$
. For any countable subset
$\{\alpha _n : n\in \mathbb{N}\}$
of
$P$
, let
$\beta =\operatorname{sup}\{\alpha _n : n\in \mathbb{N}\}$
. Then,
$\beta$
is still a countable ordinal number. Let
$\gamma =\beta +1$
. Then,
$\gamma \in \mathord {\uparrow } \gamma \in \sigma (P)$
but
$\alpha _n \not \in \mathord {\uparrow } \gamma$
for all
$n\in \mathbb{N}$
. Hence, the sequence
$(\alpha _n)_{n\in \mathbb{N}}$
does not converge to
$\gamma$
in
$\Sigma P$
. So,
$P$
is not an
$\omega$
-Rudin set by Theorem
38
. Thus,
$\Sigma P$
is not an
$\omega$
-Rudin space.
-
(h) Neither
$P_H(\Sigma P)$
nor
$(\Sigma P)^s$
is a Fréchet space by (g) and Prpposition 59. -
(i) Neither
$P_H(\Sigma P)$
nor
$(\Sigma P)^s$
is a first-countable by (h) and Proposition 57
. -
(j)
$\Sigma P$
is not second-countable by (g) and Corollary 61(1).
7. Non-Reflectivity of Category of
$\omega$
-Rudin Spaces
It is well-known that
$\mathbf{Sob}$
,
$\mathbf{Top}_w$
, and
$\mathbf{Top}_{\omega \mbox{-}w}$
are reflective in
$\mathbf{Top}_0$
(see, e.g., (Artin et al. Reference Artin, Grothendieck and Verdier1972, IV 4.2.1), (Xu et al. Reference Xu, Shen, Xi and Zhao2020a, Corollary 7.9) and (Xu et al. Reference Xu, Shen, Xi and Zhao2020b, Corollary 6.9)). So, naturally one asks the following question:
Is the category
$\boldsymbol{\omega }$
-
$\mathbf{Rud}$
of
$\omega$
-Rudin spaces reflective in
$\mathbf{Top}_{0}?$
In this section, we will show that, unlike
$\mathbf{Sob}$
,
$\mathbf{Top}_w$
, and
$\mathbf{Top}_{\omega \mbox{-}w}$
,
$\boldsymbol{\omega }$
-
$\mathbf{Rud}$
is not reflective in
$\mathbf{Top}_0$
, giving a negative answer to the about question.
In what follows,
$\mathbf{K}$
always refers to a full subcategory
$\mathbf{Top}_0$
, an object of
$\mathbf{K}$
will be called a
$\mathbf{K}$
-space.
Definition 67.
Let
$X$
be a
$T_{0}$
space. A
$\mathbf{K}$
-reflection of
$X$
is a pair
$\langle \widetilde {X},\eta _X \rangle$
consisting of a
$\mathbf{K}$
-space
$\widetilde {X}$
and a continuous mapping
$\eta _X :X\rightarrow \widetilde {X}$
satisfying that for any continuous mapping
$f:X\rightarrow Y$
to a
$\mathbf{K}$
-space, there exists a unique continuous mapping
$f^{*}:\widetilde {X}\rightarrow Y$
such that
$f^{*}\circ \eta _X =f$
, that is, the following diagram commutes.
By a standard argument,
$\mathbf{K}$
-reflections, if they exist, are unique up to homeomorphism. We shall use
$X^k$
to denote the space of the
$\mathbf{K}$
-reflection of
$X$
if it exists.
For any pair of continuous mappings
$f, g : X \rightarrow Y$
in
$\mathbf{Top}_0$
, let
$E(f, g)=\{x\in X : f(x)=g(x)\}$
and endow it with the induced topology (i.e.,
$E(f, g)$
is a subspace of
$X$
). It is easy to verify that the pair
$\langle E(f, g), i_{E(f, g)} \rangle$
is the equalizer of
$f$
and
$g$
in
$\mathbf{Top}_0$
, where
$i_{E(f, g)}$
is the inclusion of
$E(f, g)$
into
$X$
.
Definition 68.
A full subcategory
$\mathbf{K}$
of
$\mathbf{Top}_0$
is said to have equalizers if for any pair of morphisms
$f, g : X \rightarrow Y$
in
$\mathbf{K}$
, the equalizer
$E(f, g)=\{x\in X : f(x)=g(x)\}$
of
$f$
and
$g$
in
$\mathbf{Top}_0$
is a
$\mathbf{K}$
-space, or equivalently,
$E(f, g)=\{x\in X : f(x)=g(x)\}$
as a subspace of
$X$
is a
$\mathbf{K}$
-space.
As the equalizer is a special kind of the limit, from (MacLane Reference MacLane1997, page 92, Exercise 7) or (Keimel and Lawson Reference Keimel and Lawson2009, Remark 4.8), we get the following.
Lemma 69.
Let
$\mathbf{K}$
is a full subcategory
$\mathbf{Top}_0$
. If
$\mathbf{K}$
is reflective in
$\mathbf{Top}_0$
, then
$\mathbf{K}$
has equalizers.
Proposition 70.
$\boldsymbol{\omega }$
-
$\mathbf{Rud}$
does not have equalizers.
Proof.
Let
$Y$
be the space in Example 56. Then,
$Y$
is an
$\omega$
-Rudin space as shown in Example 56(f).
Define a mapping
$g: Y \rightarrow Y$
by
\begin{align*} g(y)= \begin{cases} x,& y=x\in X,\\ n+1,& y=n\in \mathbb{N}. \end{cases} \end{align*}
Then, for any
$C\in X^{(\leq \omega )}$
and
$F\in \mathbb{N}^{(\lt \omega )}$
, we have
$g^{-1}(C\cup F)=g^{-1}(C)\cup g^{-1}(F)=C\cup \{n : n+1\in F\}\in \Gamma (Y)$
by Example 56(a). Hence,
$g$
is continuous by Example 56(a) again. Clearly, the equalizer
$E(g, id_Y)=\{y\in Y : g(y)=y\}=X$
(as a subspace of
$Y$
) is the space
$X_{coc}$
, which is not an
$\omega$
-Rudin space by Example 48(d). Thus,
$\boldsymbol{\omega }$
-
$\mathbf{Rud}$
does not have equalizers.
By Lemma 69 and Proposition 70, we get the following.
Theorem 71.
$\boldsymbol{\omega }$
-
$\mathbf{Rud}$
is not reflective in
$\mathbf{Top}_0$
.
In the following, as an explicit example, we shall show that the
$\boldsymbol{\omega }$
-
$\mathbf{Rud}$
-reflection of
$X_{coc}$
(
$X$
is an uncountable set) does not exist. To do this, we first introduce the following.
Definition 72.
Let
$X$
be a topological space for which
$X$
is irreducible (i.e.,
$X\in {\mathsf{Irr}}_c(X)$
). Select a point
$\top$
such that
$\top \not \in X$
. Then,
$(\Gamma (X)\setminus \{X\})\cup \{X\cup \{\top \}\}$
(as the set of all closed sets) is a topology on
$X\cup \{\top \}$
. The resulting space is denoted by
$X_{\top }^{\flat }$
.
Remark 73. Let
$X$
be a topological space. Select a point
$\top$
such that
$\top \not \in X$
. Then,
$\Gamma (X)\cup \{X\cup \{\top \}\}$
(as the set of all closed sets) is a topology on
$X\cup \{\top \}$
. The resulting space is denoted by
$X_{\top }$
.
-
(1) If
$X$
is not irreducible, then there exist
$A, B\in \Gamma (X)\setminus \{X\}$
such that
$X=A\cup B$
, whence
$(\Gamma (X)\setminus \{X\})\cup \{X\cup \{\top \}\}$
is not a topology on
$X\cup \{\top \}$
. -
(2) When
$X\in {\mathsf{Irr}}_c(X)$
, we have
$X\not \in \Gamma (X_{\top }^{\flat })$
but
$X\in \Gamma (X_{\top })$
, and hence
$\Gamma (X_{\top }^{\flat })$
is strictly coarser than
$\Gamma (X_{\top })$
.
Lemma 74 (Shan et al. Reference Shan, Bao, Wen and Xu2022, Corollary 6.9). Let
$\mathbf{K}$
be a full subcategory of
$\mathbf{Top}_0$
, which is closed with respect to homeomorphisms and contains
$\mathbf{Sob}$
. For a
$T_0$
-space
$X$
, which is not a
$\mathbf{K}$
-space, if
$\mathbf{Irr}_{c}(X)=\{\overline {\{x\}} : x\in X\}\cup \{X\}$
, then we have the following two conclusions:
-
(1)
$\langle X_{\top }^{\flat },\eta _{X}^{\flat } \rangle$
is a sobrification of
$X$
, where
$\eta _{\top }^{\flat }: X\rightarrow X_{\top }^{\flat }$
is defined by
$X_{\top }^{\flat }(x)=x$
for each
$x\in X$
. -
(2) If the sobrification
$X^s$
(and hence any sobrification) is not a
$\mathbf{K}$
-reflection of
$X$
, then the
$\mathbf{K}$
-reflection of
$X$
does not exist.
Now, we show that the
$\boldsymbol{\omega }$
-
$\mathbf{Rud}$
-reflection of
$X_{coc}$
(
$X$
is an uncountable set) does not exist.
Theorem 75.
Let
$X$
be an uncountable set. Then, the
$\boldsymbol{\omega }$
-
$\mathbf{Rud}$
-reflection of
$X_{coc}$
does not exist. So,
$\boldsymbol{\omega }$
-
$\mathbf{Rud}$
is not reflective in
$\mathbf{Top}$
.
Proof.
Clearly,
$\boldsymbol{\omega }$
-
$\mathbf{Rud}$
is closed with respect to homeomorphisms, and by Theorem 30 it contains
$\mathbf{Sob}$
. By Example 48,
$X_{coc}$
is not an
$\omega$
-Rudin space and
${\mathsf{Irr}}_c(X_{coc})=\{\{x\} : x\in X\}\cup \{X\}$
, and hence,
$\langle (X_{coc})_{\top }^{\flat },\eta _{X_{coc}}^{\flat } \rangle$
is a sobrification of
$X_{coc}$
by Lemma 74. Now, we show that the sobrification
$\langle (X_{coc})_{\top }^{\flat },\eta _{X_{coc}}^{\flat } \rangle$
is not an
$\boldsymbol{\omega }$
-
$\mathbf{Rud}$
-reflection of
$X_{coc}$
.
Assume, on the contrary, that
$\langle (X_{coc})_{\top }^{\flat },\eta _{X_{coc}}^{\flat } \rangle$
is an
$\boldsymbol{\omega }$
-
$\mathbf{Rud}$
-reflection of
$X_{coc}$
. Let
$Y=X\cup \mathbb{N}$
and endow
$Y$
with the topology
$\{\emptyset , Y\}\bigcup \{U\cup V: U\in \mathscr O(X_{coc}), V\in \mathscr O({\mathbb{N}_{cof}})\}$
. Then, by Example 56(f), the space
$Y$
is an
$\omega$
-Rudin space. Consider the inclusion embedding
$i_{X_{coc}} : X_{coc}\longrightarrow Y$
. Then, there is a unique continuous mapping
$(i_{X_{coc}})^* : (X _{coc})_{\top }^{\flat } \rightarrow Y$
such that
$(i_{X_{coc}})^*\circ \eta _{X_{coc}}^{\flat }=i_{X_{coc}}$
, that is, the following diagram commutes.
Then, for each
$x\in X$
,
$(i_{X_{coc}})^*(x)=(i_{X_{coc}})^*(\eta _{X_{coc}}^{\flat }(x))=i_{X_{coc}}(x)=x$
. For any
$x\in X$
, by the continuity of
$(i_{X_{coc}})^*$
and
$x\in X\cup \{\top \}=\textrm {cl}_{X_{coc}}\{\top \}$
, we have
$x=(i_{X_{coc}})^*(x)\in (i_{X_{coc}})^*(\textrm {cl}_{X_{coc}}\{\top \})\subseteq \textrm {cl}_Y\{(i_{X_{coc}})^*(\top )\}=\{(i_{X_{coc}})^*(\top )\}$
(note that
$Y$
is
$T_1$
), and hence
$x=(i_{X_{coc}})^*(\top )$
, a contradiction.
Therefore, the sobrification
$\langle (X_{coc})_{\top }^{\flat },\eta _{X_{coc}}^{\flat } \rangle$
is not an
$\boldsymbol{\omega }$
-
$\mathbf{Rud}$
-reflection of
$X_{coc}$
. By Lemma 74, the
$\boldsymbol{\omega }$
-
$\mathbf{Rud}$
-reflection of
$X_{coc}$
does not exist.
As the space
$Y$
in the proof of Theorem 75 is an
$\omega$
-Rudin space, it is an Rudin space, and hence, a
$\mathsf{WD}$
space by Lemma 17. It follows from Lemma 22 that
$Y$
is an
$\omega$
-
$\mathsf{WD}$
space. Therefore, from the proof of Theorem 75 and Lemma 74, we get the following.
Proposition 76.
None of
$\mathbf{Rud}$
,
$\mathbf{WD}$
, and
$\boldsymbol{\omega }$
-
$\mathbf{WD}$
is reflective in
$\mathbf{Top}_0$
.
8. Smyth Power Spaces of
$\omega$
-Rudin Spaces
The Smyth power construction is an important structure in domain theory and non-Hausdorff topology, which play a fundamental role in modeling the semantics of nondeterministic programming languages (see Abramsky and Jung (Reference Abramsky and Jung1994); Gierz et al. (Reference Gierz, Hofmann, Keimel, Lawson, Mislove and Scott2003); Goubault (Reference Goubault-Larrecq2013); Heckmann (Reference Heckmann1990,Reference Heckmann1992); Smyth (Reference Smyth1978); Schalk (Reference Schalk1993)). There naturally arises a question of which topological properties are preserved by the Smyth power construction.
For the sobriety and the well-filteredness, we have the following two important results.
Theorem 77 (Heckmann-Keimel-Schalk Theorem) (Heckmann and Keimel Reference Heckmann and Keimel2013, Theorem 3.13) (Schalk Reference Schalk1993, Lemma 7.20). For a
$T_0$
-space
$X$
, the following conditions are equivalent:
-
(1)
$X$
is sober.
-
(2) For any
$\mathscr A\in {\mathsf{Irr}}(P_S(X))$
and
$U\in \mathscr O(X)$
,
$\bigcap \mathscr A\subseteq U$
implies
$K \subseteq U$
for some
$K\in \mathscr A$
. -
(3)
$P_S(X)$
is sober.
Theorem 78 (Xu et al. Reference Xu, Shen, Xi and Zhao2020b, Theorem 5.3) (Xu et al. Reference Xu, Xi and Zhao2021b, Theorem 4). For a
$T_0$
-space, the following conditions are equivalent:
-
(1)
$X$
is well-filtered (resp.,
$\omega$
-well-filtered).
-
(2)
$P_S(X)$
is a
$d$
-space (resp., an
$\omega$
-
$d$
-space).
-
(3)
$P_S(X)$
is well-filtered (
$\omega$
-well-filtered).
Now, we consider the property of being an
$\omega$
-Rudin space.
Lemma 79 (Xu et al. Reference Xu, Miao and Li2025, Lemma 4.5). For a
$T_0$
-space
$X$
and
$A, B\subseteq X$
, we have the following conclusions:
-
(1)
$\downarrow _{\mathsf{K}(X)} \xi _X(A)=\Diamond _{\mathsf{K}(X)} A=\Diamond _{\mathsf{K}(X)} \downarrow A$
. -
(2)
$\downarrow _{\mathsf{K}(X)} \xi _X(A)=\downarrow _{\mathsf{K}(X)} \xi _X(B)$
iff
$\downarrow A=\downarrow B$
. -
(3)
$\Diamond _{\mathsf{K}(X)} A=\Diamond _{\mathsf{K}(X)} B$
iff
$\downarrow A=\downarrow B$
. -
(4)
$\textrm {cl}_{P_S(X)}\xi _X(A)=\Diamond _{\mathsf{K}(X)} \overline {A}$
. -
(5)
$\textrm {cl}_{P_S(X)}\Diamond _{\mathsf{K}(X)} A=\Diamond _{\mathsf{K}(X)} \overline {A}$
.
Proposition 80.
Let
$X$
be a
$T_0$
-space. Consider the following two conditions:
-
(1) The Smyth power space
$P_S(X)$
is
$\omega$
-Rudin.
-
(2)
$X$
is an
$\omega$
-Rudin space.
Then, (1)
$\Rightarrow$
(2), and two conditions are equivalent if
$X$
is
$\omega$
-well-filtered.
Proof.
(1)
$\Rightarrow$
(2): Let
$A\in {\mathsf{Irr}}_c(X)$
. Then, by the continuity of
$\xi _X : X \rightarrow P_S(X)$
, Lemmas 3 and 79, we have
$\Diamond _{\mathsf{K}(X)} A=\textrm {cl}_{P_S(X)}\xi _X(A)\in {\mathsf{Irr}}_c(P_S(X))$
. As
$P_S(X)$
is an
$\omega$
-Rudin space, by Theorem 38, there is
$\{K_n : n\in \mathbb{N}\}\subseteq \Diamond _{\mathsf{K}(X)} A$
such that
$(K_n)_{n\in \mathbb{N}}$
simultaneously converges to all points of
$\Diamond _{\mathsf{K}(X)} A$
in
$P_S(X)$
. For each
$n\in \mathbb{N}$
, by
$K_n\in \Diamond _{\mathsf{K}(X)} A$
, we can choose a point
$a_n\in K_n\cap A$
. Then,
$\{a_n : n\in \mathbb{N}\}\subseteq A$
. Now, we show that the sequence
$(a_n)_{n\in \mathbb{N}}$
simultaneously converges to all points of
$A$
. For any
$a\in A$
and
$U\in \mathscr O(X)$
with
$a\in U$
, we have
$\mathord {\uparrow } a\in \Diamond _{\mathsf{K}(X)} A$
and
$\mathord {\uparrow } a\in \Box U\in \mathscr O(P_S(X))$
, whence
$(K_n)_{n\in \mathbb{N}}$
converges to
$\mathord {\uparrow } a$
in
$P_S(X)$
. So, there is
$n^\prime \in \mathbb{N}$
such that
$K_n\in \Box U$
for all
$n\geq n^\prime$
. Then,
$a_n\in K_n\subseteq U$
for all
$n\geq n^\prime$
. So,
$(a_n)_{n\in \mathbb{N}}$
simultaneously converges to all points of
$A$
. Thus,
$X$
is
$\omega$
-Rudin.
(2)
$\Rightarrow$
(1): Suppose that
$X$
is both an
$\omega$
-Rudin space and an
$\omega$
-well-filtered space. Then, by Theorem 30,
$X$
is sober. By Theorem 77,
$P_S(X)$
is a sober space and hence by Theorem 30 is an
$\omega$
-Rudin space.
But we do not know whether a similar result to Theorems 77 and 78 holds for
$\omega$
-Rudin spaces. So, naturally we pose the following question.
Question 81.
For an
$\omega$
-Rudin space
$X$
, is its Smyth power space
$P_S(X)$
an
$\omega$
-Rudin space?
Acknowledgements
The author would like to thank the referee for the numerous and very helpful suggestions that have improved this paper substantially.
