1. Introduction
Function spaces (equipped with certain topologies) are important structures in topology and domain theory (see Engelking Reference Engelking1989; Gierz et al. Reference Gierz, Hofmann, Keimel, Lawson, Mislove and Scott2003), which was initially introduced by Dana Scott (Scott Reference Scott1970). As a special kind of mathematical structures, domains serve as mathematical universes within which people can interpret higher-order functional programming languages, and cartesian closed categories of domains (more generally, certain topological spaces) are appropriate for models of various typed and untyped lambda-calculi and functional programming languages (see Gierz et al. Reference Gierz, Hofmann, Keimel, Lawson, Mislove and Scott2003). Since whether certain properties of topological spaces are preserved when passing to function spaces is connected with the cartesian closed category of topological spaces, this question has attracted considerable attention in domain theory and non-Hausdorff topology, especially for domains (which are a special kind of topological spaces when endowed with the Scott topology), sober spaces, d-spaces and well-filtered spaces (see Ershov et al. 2020; Gierz et al. Reference Gierz, Hofmann, Keimel, Lawson, Mislove and Scott2003; Liu et al. Reference Liu, Li and Ho2021; Xu Reference Xu2021).
There exists a quite satisfactory theory which deals with the cartesian closedness of domains. Jung (Reference Jung1989, Reference Jung1990), Plotkin (Reference Plotkin1976) and Smyth (Reference Smyth1983) have made essential contributions to this theory. For topological spaces, it is well-known that if X is a
$T_0$
space and Y a sober space, then the function space
$\mathbb{C}(X, Y)$
of all continuous functions
$f:X\rightarrow Y$
equipped with the topology of pointwise convergence is sober (see, for example, Gierz et al. Reference Gierz, Hofmann, Keimel, Lawson, Mislove and Scott2003, Exercise O-5.16). Furthermore, in Ershov et al. (2020), it was shown that for any
$T_{0}$
space X, a
$T_0$
space Y is a d-space (resp., sober space) iff the function space
$\mathbb{C}(X, Y)$
equipped with the topology of pointwise convergence is a d-space (resp., sober space). It is known that for a
$T_0$
space X and a d-space Y, the function space
$\mathbb{C}(X, Y)$
equipped with the Isbell topology is a d-space (cf. Gierz et al. Reference Gierz, Hofmann, Keimel, Lawson, Mislove and Scott2003, Lemma II-4.3). Conversely, in Liu et al. (Reference Liu, Li and Ho2021), the authors showed that if the function space
$\mathbb{C}(X, Y)$
equipped with the Isbell topology is a d-space, then Y is a d-space. For the well-filteredness, it was proved in Liu et al. (Reference Liu, Li and Ho2021) that for any core-compact space X and well-filtered space Y, the function space
$\mathbb{C}(X, Y)$
equipped with the Isbell topology is well-filtered.
In order to provide a uniform approach to d-spaces, sober spaces and well-filtered spaces, Xu (Reference Xu2021) introduced the concepts of irreducible subset system H and H-sober spaces and developed a general framework for dealing with all these spaces. The irreducible subset systems
$\mathcal D$
,
$\mathcal R$
,
$\mathsf{WD}$
and
$\mathsf{RD}$
are four important ones, where for each
$T_0$
space X,
$\mathcal D(X)$
is the set of all directed subsets of X,
$\mathcal R(X)$
is the set of all irreducible subsets of X,
$\mathsf{WD}(X)$
is the set of all well-filtered determined subsets of X, and
$\mathsf{RD}(X)$
is the set of all Rudin subsets of X (see Xu Reference Xu2021; Xu et al. Reference Xu, Shen, Xi and Zhao2020). So the d-spaces, sober spaces and well-filtered spaces are three special types of H-sober spaces. It was proved in Xu (Reference Xu2021) that for a
$T_0$
space X and an H-sober space Y, the function space
$\mathbb{C}(X, Y)$
equipped with the topology of pointwise convergence is H-sober. One immediate corollary is that for a
$T_0$
space X and a sober space (resp., d-space, well-filtered space) Y, the function space
$\mathbb{C}(X, Y)$
equipped with the topology of pointwise convergence is a sober space (resp., d-space, well-filtered space).
In this paper, we mainly study the function spaces related to H-sober spaces, especially to d-spaces, well-filtered spaces and sober spaces. The paper is organized as follows:
In Section 2, some fundamental concepts and notation are introduced which will be used in the whole paper.
In Section 3, we briefly recall some basic concepts and results about irreducible subset systems and H-sober spaces that will be used in the other sections.
In Section 4, for the convenience of discussions of function spaces equipped with the pointwise convergence topology, compact-open topology, Isbell topology and Scott topology, several basic lemmas are given.
In Section 5, we mainly discuss the function spaces equipped with Isbell topology. For an irreducible subset system H and
$T_{0}$
spaces X and Y, the interlinks between the H-sobriety of Scott space
$\Sigma \mathcal O(X)$
, the H-sobriety of Y and the H-sobriety of the function space
$\mathbb{C}(X, Y)$
equipped with the Isbell topology are discussed. It is proved that the following three conditions are equivalent: (1)
$\Sigma \mathcal O(X)$
is H-sober; (2) for every H-sober space Y, the function space
$\mathbb{C}(X, Y)$
equipped with the Isbell topology is H-sober; (3) for every H-sober space Y, the Isbell topology on
$\mathbb{C}(X, Y)$
has property S with respect to H. As an immediate corollary, we get that for a
$T_{0}$
space X, Y is a d-space (resp., well-filtered space) iff the function space
$\mathbb{C}(X, Y)$
equipped with the Isbell topology is a d-space (resp., well-filtered space). It is shown that for any
$T_0$
space X for which the Scott space
$\Sigma \mathcal O(X)$
is non-sober, the function space
$\mathbb{C}(X, \Sigma 2)$
equipped with the Isbell topology is not sober.
In Sections 6 and 7, we investigate the function spaces equipped with the pointwise convergence topology and compact-open topology, and the Scott topology, respectively.
2. Preliminaries
In this section, we briefly recall some fundamental concepts and notation that will be used in this paper; more details can be founded in Engelking (Reference Engelking1989), Gierz et al. (Reference Gierz, Hofmann, Keimel, Lawson, Mislove and Scott2003), Goubault-Larrecq (2013).
For a poset P and
$A \subseteq P$
, let
$\mathord{\uparrow} A=\{x\in P: a\leq x \mbox{ for some }a\in A \}$
(dually
$\mathord{\downarrow} A=\{x \in P: x \leq a \mbox{ for some }a\in A\})$
. For
$A=\{x\}$
,
$\mathord{\uparrow} A$
and
$\mathord{\downarrow} A$
are shortly denoted by
$\mathord{\uparrow} x$
and
$\mathord{\downarrow} x,$
respectively. A subset A is called a lower set (resp., an upper set) if
$A=\mathord{\downarrow} A$
(resp.,
$A=\mathord{\uparrow} A$
). Let
$P^{(<\omega)}=\{F\subseteq P : F \mbox{ is a finite set}\}$
and
$\mathbf{Fin} P=\{\uparrow F : F\in P^{(<\omega)}\setminus \{\emptyset\}\}$
. A subset D of P is directed provided that it is non-empty, and every finite subset of D has an upper bound in D. The set of all directed sets of P is denoted by
$\mathcal{D}(P)$
. P is said to be a directed complete poset, a dcpo for short, if every directed subset of P has the least upper bound in P. As in Gierz et al. (Reference Gierz, Hofmann, Keimel, Lawson, Mislove and Scott2003), the upper topology on a poset P, generated by the complements of the principal ideals of P, is denoted by
$\upsilon (P)$
. The upper sets of P form the (upper) Alexandroff topology
$\gamma (P)$
. The space
$\Gamma \!\!P=(P, \gamma(P))$
is called the Alexandroff space of P. A subset U of a poset P is called Scott open if
$U=\mathord{\uparrow} U$
and
$D\cap U\neq\emptyset$
for all directed sets
$D\subseteq P$
with
$\vee D\in U$
whenever
$\vee D$
exists. The topology formed by all Scott open sets of P is called the Scott topology, written as
$\sigma(P)$
.
$\Sigma P=(P, \!\sigma(P))$
is called the Scott space of P. Clearly,
$\Sigma P$
is a
$T_0$
space.
The following result is well-known (cf. Gierz et al. Reference Gierz, Hofmann, Keimel, Lawson, Mislove and Scott2003).
Lemma 1. Let P, Q be posets and
$f : P \rightarrow Q$
. Then, the following two conditions are equivalent:
-
(1) f is Scott continuous, that is,
$f : \Sigma P \rightarrow \Sigma Q$
is continuous. -
(2) For any
$D\in \mathcal D(P)$
for which
$\vee D$
exists,
$f(\vee D)=\vee f(D)$
.
Given a
$T_0$
space X, we can define a partial order
$\leq_{X}$
, called the specialization order, which is defined by
$x \leq_X y$
iff
$x \in \overline{\{y\}}$
. Let
$\Omega X$
denote the poset
$(X, \leq_X)$
. Clearly, each open set is an upper set, and each closed set is a lower set with respect to the partial order
$\leq_{X}$
. Unless otherwise stated, throughout the paper, whenever an order-theoretic concept is mentioned in a
$T_0$
space, it is to be interpreted with respect to the specialization order. We often use X itself to denote the poset
$\Omega X$
. Let
$\mathcal{O}(X)$
(resp.,
$\mathcal{C}(X)$
) be the set of all open subsets (resp., closed subsets) of X and denote
$\mathcal{S}(X)=\{\{x\}: x\in X\}$
and
$\mathcal{D}(X)=\{D\subseteq X: D$
is a directed set of
$X\}$
. Let
$\mathcal{S}_c(X)=\{\overline {\{x\}}: x\in X\}$
and
$\mathcal{D}_c(X)=\{\overline D : D\in \mathcal{D}(X)\}$
. A
$T_0$
topology
$\tau$
on a poset P is called order compatible if the specialization order agrees with the original order on P or, equivalently,
$\upsilon (P)\subseteq \tau\subseteq \gamma (P)$
.
Lemma 2. (Keimel et al. Reference Keimel and Lawson2009, Lemma 6.2) Let
$f : X \rightarrow Y$
be a continuous mapping of
$T_0$
spaces. If
$D\in \mathcal {D}(X)$
has a supremum to which it converges, then f(D) is directed and has a supremum in Y to which it converges, and
$f(\vee D)=\vee f(D)$
.
Corollary 3. Let P be a poset and Y a
$T_0$
space. If
$f : \Sigma P \rightarrow Y$
is continuous, then
$f : \Sigma P \rightarrow \Sigma Y$
is continuous.
A non-empty subset A of a
$T_0$
space X is called irreducible if for any
$\{F_1, F_2\}\subseteq \mathcal{C}(X)$
,
$A\subseteq F_1\cup F_2$
implies
$A\subseteq F_1$
or
$A\subseteq F_2$
. We denote by
${\mathsf{Irr}}(X)$
(resp.,
${\mathsf{Irr}}_c(X)$
) the set of all irreducible (resp., irreducible closed) subsets of X. Clearly, every subset of X that is directed under
$\leq_{X}$
is irreducible. A topological space X is called sober, if for any
$ F\in{\mathsf{Irr}}_c(X)$
, there is a unique point
$x\in X$
such that
$F=\overline{\{x\}}$
.
The category of all sets and mappings is denoted by
$\mathbf{Set}$
and the category of all
$T_0$
spaces with continuous mappings is denoted by
$\mathbf{Top}_0$
. A
$T_0$
space X is called a d-space (or monotone convergence space) if X (with the specialization order) is a dcpo and
$\mathcal O(X) \subseteq \sigma(X)$
(cf. Gierz et al. Reference Gierz, Hofmann, Keimel, Lawson, Mislove and Scott2003). Clearly, for a dcpo P,
$\Sigma \!\! P$
is a d-space.
One can directly get the following result (cf. Xu et al. Reference Xu, Shen, Xi and Zhao2020, Proposition 3.3).
Proposition 4. For a
$T_0$
space X, the following conditions are equivalent:
-
(1) X is a d-space.
-
(2)
$\mathcal D_c(X)=\mathcal S_c(X)$
, that is, for each
$D\in \mathcal D(X)$
, there exists a (unique) point
$x\in X$
such that
$\overline{D}=\overline{\{x\}}$
.
For a
$T_0$
space X, let
$2^{X}$
be the set of all subsets of X. A subset A of X is called saturated if A equals the intersection of all open sets containing it (equivalently, A is an upper set in the specialization order). We denote by
$\mathord{\mathsf{K}}(X)$
the poset of non-empty compact saturated subsets of X with the Smyth preorder, i.e., for
$K_{1}, K_{2}\in \mathord{\mathsf{K}}(X)$
,
$K_{1}\sqsubseteq K_{2}$
iff
$K_{2}\subseteq K_{1}$
. For
$G\subseteq X$
, let
$\Phi (G)=\{U\in \mathcal O(X) : G\subseteq U\}$
. We denote by
$\phi (\mathcal O(X))$
the topology on
$\mathcal O(X)$
generated by
$\{\Phi (C) : C \hbox{ is a compact subset of } X\}$
(as a base). It is easy to verify that
$\upsilon(\mathcal O(X))\subseteq \phi (\mathcal O(X))\subseteq \sigma(\mathcal O(X))$
and the family
$\{\Phi(K) : K\in \mathsf{K}(X)\cup\{\emptyset\}\}$
is also a base of
$\phi (\mathcal O(X))$
. The space X is called well-filtered, if for any open set U and any filtered family
$\mathcal{K}\subseteq \mathord{\mathsf{K}}(X)$
,
$\bigcap\mathcal{K}{\subseteq} U$
implies
$K{\subseteq} U$
for some
$K{\in}\mathcal{K}$
.
In Xi et al. (Reference Lu and Li2017), the following useful result was given.
Proposition 5. (Xi et al. Reference Lu and Li2017, Corollary 3.2) If L is a complete lattice, then
$\Sigma L$
is well-filtered.
For a dcpo P and
$x, y\in P$
, we say x is way below y, written
$x\ll y$
, if for each
$D\in \mathcal D(P)$
,
$y\leq\vee D$
implies
$x\leq d$
for some
$d\in D$
. Let
$\Downarrow x = \{u\in P : u\ll x\}$
. P is called a continuous domain, if for each
$x\in P$
,
$\Downarrow x$
is directed and
$x=\vee\Downarrow x$
. When a complete lattice L is continuous, we call L a continuous lattice. A topological space X is called core-compact if
$\mathcal O(X)$
is a continuous lattice.
3. Irreducible Subset Systems and H-Sober Spaces
In this section, we briefly recall some basic concepts and results about irreducible subset systems and H-sober spaces that will be used in the other sections. For further details, we refer the reader to Shen et al. (Reference Shen, Xi, Xu and Zhao2019), Xu (Reference Xu2021), Xu et al. (Reference Xu, Shen, Xi and Zhao2020). In what follows, all topological spaces will be supposed to be non-empty spaces.
For a
$T_0$
space X and
$\mathcal{K}\subseteq \mathord{\mathsf{K}}(X)$
, let
$M(\mathcal{K})=\{A\in \mathcal{C}(X): K\cap A\neq\emptyset \mbox{ for all } K\in \mathcal{K}\}$
and
$m(\mathcal{K})=\{A\in \mathcal{C}(X): A \mbox{ is a minimal member of } M(\mathcal{K})\}$
.
Definition 6. (Shen et al. Reference Shen, Xi, Xu and Zhao2019; Xu et al. Reference Xu, Shen, Xi and Zhao2020) Let X be a
$T_0$
space and A a non-empty subset of X.
-
(1) A is said to have the Rudin property (which is called compactly filtered property in Reference Shen, Xi, Xu and ZhaoShen et al. 2019), if there exists a filtered family
$\mathcal K\subseteq \mathord{\mathsf{K}}(X)$
such that
$\overline{A}\in m(\mathcal{K})$
(that is,
$\overline{A}$
is a minimal closed set that intersects all members of
$\mathcal{K}$
). Let
$\mathsf{RD}(X)=\{A\subseteq X: A \mbox{{ has Rudin property}}\}$
and
$\mathsf{RD}_{c}(X)=\{\overline{A}: A \in \mathsf{RD}(X)\}$
. The sets in
$\mathsf{RD}(X)$
will also be called Rudin sets. -
(2) A is called a well-filtered determined set (WD-set for short) if for any continuous mapping
$f:X\rightarrow Y$
into a well-filtered space Y, there exists a unique
$y_{A}\in Y$
such that
$\overline{f(A)}=\overline{\{y_{A}\}}$
. Denote by
$\mathsf{WD}(X)$
the set of all well-filtered determined subsets of X and
$\mathsf{WD}_{c}(X)=\{\overline{A}: A\in \mathsf{WD}(X)\}$
. Clearly, a subset A of a space X is well-filtered determined iff
$\overline{A}$
is well-filtered determined.
Lemma 7. (Xu et al. Reference Xu, Shen, Xi and Zhao2020, Proposition 6.2) Let X be a
$T_{0}$
space. Then,
$\mathcal{S}(X)\subseteq \mathcal{D}(X)\subseteq \mathsf{RD}(X)\subseteq \mathsf{WD}(X)\subseteq {\mathsf{Irr}}(X)$
.
Lemma 8. (Reference Shen, Xi, Xu and ZhaoShen et al. 2019, Lemma 2.5; Reference Xu, Shen, Xi and ZhaoXu et al. 2020, Lemma 6.23) Let X, Y be two
$T_{0}$
spaces and
$f:X\rightarrow Y$
a continuous mapping. If
$A\in \mathsf{RD}(X)$
(resp.,
$A\in \mathsf{WD}(X)$
), then
$f(A)\in \mathsf{RD}(Y)$
(resp.,
$f(A)\in \mathsf{WD}(Y)$
).
Using Rudin sets and WD-sets, we have the following characterization of well-filtered spaces.
Proposition 9. (Xu et al. Reference Xu, Shen, Xi and Zhao2020, Corollary 7.11) Let X be a
$T_{0}$
space. Then, the following conditions are equivalent:
-
(1) X is well-filtered.
-
(2)
$\mathsf{WD}_{c}(X)=\mathcal{S}_{c}(X)$
, that is, for any
$A\in \mathsf{WD}(X)$
, there exists a unique
$a\in X$
such that
$\overline{A}=\overline{\{a\}}$
. -
(3)
$\mathsf{RD}_{c}(X)=\mathcal{S}_{c}(X)$
, that is, for any
$A\in \mathsf{RD}(X)$
, there exists a unique
$a\in X$
such that
$\overline{A}=\overline{\{a\}}$
.
In order to provide a uniform approach to d-spaces, sober spaces and well-filtered spaces and develop a general framework for dealing with all these spaces, Xu (Reference Xu2021) introduced the following concepts.
Definition 10. (Xu Reference Xu2021) (1) A covariant functor
${\rm H}: \mathbf{Top}_0 \rightarrow \mathbf{Set}$
is called a subset system on
$\mathbf{Top}_0$
provided that the following two conditions are satisfied:
-
(1)
$\mathcal S(X)\subseteq {\rm H}(X)\subseteq 2^X$
(the set of all subsets of X) for each
$X\in$
ob(
$\mathbf{Top}_0$
). -
(2) For any continuous mapping
$f: X \rightarrow Y$
in
$\mathbf{Top}_0$
,
${\rm H}(f)(A)=f(A)\in {\rm H}(Y)$
for all
$A\in {\rm H}(X)$
.
(2) A subset system
${\rm H}: \mathbf{Top}_0 \rightarrow \mathbf{Set}$
is called an irreducible subset system, or an R-subset system for short, if
${\rm H}(X)\subseteq {\mathsf{Irr}} (X)$
for all
$X\in$
ob(
$\mathbf{Top}_0$
).
In what follows, the capital letter H always stands for an R-subset system
${\rm H} : \mathbf{Top}_0 \rightarrow \mathbf{Set}$
. For a
$T_0$
space X, let
${\rm H}_c(X)=\{\overline{A} : A\in {\rm H}(X)\}$
. Define a partial order
$\leq$
on the set of all R-subset systems by
${\rm H}_1\leq {\rm H}_2$
iff
$ {\rm H}_1(X)\subseteq {\rm H}_2(X)$
for all
$T_0$
spaces X.
Here are some important examples of R-subset systems used in this paper:
-
(1)
$\mathcal S$
(for
$X\in$
ob(
$\mathbf{Top}_0$
),
$\mathcal S(X)$
is the set of all single point subsets of X). -
(2)
$\mathcal D$
(for
$X\in$
ob(
$\mathbf{Top}_0$
),
$\mathcal D(X)$
is the set of all directed subsets of X). -
(3)
$\mathcal R$
(for
$X\in$
ob(
$\mathbf{Top}_0$
),
$\mathcal R(X)$
is the set of all irreducible subsets of X). -
(4)
$\mathsf{RD}$
(for
$X\in$
ob(
$\mathbf{Top}_0$
),
$\mathsf{RD}(X)$
is the set of all Rudin subsets of X). -
(5)
$\mathsf{WD}$
(for
$X\in$
ob(
$\mathbf{Top}_0$
),
$\mathsf{WD}(X)$
is the set of all well-filtered determined subsets of X).
By Lemmas 7 and 8,
$\mathsf{WD}$
and
$\mathsf{RD}$
are R-subset systems, and
$\mathcal S\leq \mathcal D\leq \mathsf{RD}\leq \mathsf{WD}\leq \mathcal R$
.
Definition 11. (Xu Reference Xu2021) Let
${\rm H} : \mathbf{Top}_0 \rightarrow \mathbf{Set}$
be an R-subset system and X a T
0 space. X is called H-sober if for any
$A\in {\rm H}(X)$
, there is a (unique) point
$x\in X$
such that
$\overline{A}=\overline{\{x\}}$
or, equivalently, if
${\rm H}_c(X)=\mathcal S_c(X)$
.
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
$f\circ g=id_{Y}$
.
Lemma 12. (Xu Reference Xu2021, Proposition 4.26) A retract of an H-sober space is H-sober.
Corollary 13. A retract of a sober space (resp., d-space, well-filtered space) is a sober space (resp., d-space, well-filtered space).
4. Some Basic Lemmas
For
$T_0$
spaces X and Y, there are four important topologies on the set
$\mathbb{C}(X, Y)$
of all continuous functions from X to Y, namely the pointwise convergence topology, compact-open topology, Isbell topology and Scott topology. In order to discuss the function spaces equipped with these four topologies, we shall give some basic lemmas in this section.
First, we recall the definitions of pointwise convergence topology, Isbell topology and compact-open topology. For further details, we refer the reader to Engelking (Reference Engelking1989), Gierz et al. (Reference Gierz, Hofmann, Keimel, Lawson, Mislove and Scott2003) and Goubault-Larrecq (2013). In what follows,
$\mathbb{C}(X, Y)$
always means the set of all continuous functions from X to Y.
Definition 14. Let X and Y be topological spaces.
-
(1) For a point
$x\in X$
and an open set
$V\in \mathcal{O}(Y)$
, let
$S(x, V)=\{f\in \mathbb{C}(X, Y) : f(x) \in V\}$
. The set
$\{S(x, V): x\in X, V\in \mathcal{O}(Y)\}$
is a subbasis for the pointwise convergence topology (i.e., the relative product topology) on
$\mathbb{C}(X, Y)$
. Let
$[X\rightarrow Y]_{P}$
denote the function space
$\mathbb{C}(X, Y)$
endowed with the topology of pointwise convergence. -
(2) The compact-open topology on the set
$\mathbb{C}(X, Y)$
is generated by the subsets of the form
$N(K\rightarrow V)=\{f\in \mathbb{C}(X,Y): f(K)\subseteq V\}$
, where K is compact in X and V is open in Y. Let
$[X\rightarrow Y]_{K}$
denote the function space
$\mathbb{C}(X, Y)$
endowed with the compact-open topology. -
(3) The Isbell topology on the set
$\mathbb{C}(X, Y)$
is generated by the subsets of the form
$N(\mathcal H\leftarrow V)=\{f\in \mathbb{C}(X,Y):f^{-1}(V)\in \mathcal H\},$
where
$\mathcal H$
is a Scott open subset of the complete lattice
$\mathcal{O}(X)$
and V is open in Y. Let
$[X\rightarrow Y]_{I}$
denote the function space
$\mathbb{C}(X, Y)$
endowed with the Isbell topology.
For a topological space X and a
$T_0$
space Y, Y (with the specialization order) is a poset, whence
$\mathbb{C}(X, Y)$
is a poset with the pointwise order. Denote by
$[X\rightarrow Y]_{\Sigma}$
the Scott space
$\Sigma \mathbb{C}(X, Y)$
. It is easy to see that the pointwise convergence topology, the compact-open topology and the Isbell topology on
$\mathbb{C}(X, Y)$
are all order compatible, and for
$x\in X$
,
$K\in \mathsf{K}(X)$
and
$V\in \mathcal O(Y)$
, we clearly have that
$S(x, V)=N(\{x\}\rightarrow V)$
and
$N(K\rightarrow V)=N(\mathcal H_K\leftarrow V)$
, where
$\mathcal H_K=\{U\in \mathcal O(X) : K\subseteq U\}\in \sigma (\mathcal O(X))$
. So we have the following result.
Lemma 15. (Gierz et al. Reference Gierz, Hofmann, Keimel, Lawson, Mislove and Scott2003, Lemma II-4.2) Let X, Y be
$T_0$
spaces. Then,
$$\upsilon(\mathbb{C}(X, Y))\subseteq \mathcal O([X\rightarrow Y]_{P})\subseteq \mathcal O([X\rightarrow Y]_{K})\subseteq \mathcal O([X\rightarrow Y]_{I})\subseteq \gamma (\mathbb{C}(X, Y)).$$
If X is a core-compact sober space, then the Isbell topology and the compact-open topology agree.
Lemma 16. Let X, Y be
$T_0$
spaces and
$x\in X$
. Then, the evaluation function at x
$$E^{P}_x:[X\rightarrow Y]_{P}\rightarrow Y, f\mapsto \! f(x),$$
is continuous. Therefore, if
$\mathcal T$
is a topology on
$\mathbb{C}(X, Y)$
which is finer than the pointwise convergence topology, then the function
$E^{\mathcal T}_x: (\mathbb{C}(X, Y), \mathcal T) \rightarrow Y, f\mapsto \! f(x)$
is continuous.
Proof. For any
$V\in \mathcal O(Y)$
, we have
$(E^{P}_x)^{-1}(V)=\{f\in \mathbb{C}(X, Y): f(x)\in V\}=S(x, V)$
, which is open in
$[X\rightarrow Y]_{P}$
. So
$E^{P}_x$
is continuous. 
By Lemmas 15 and 16, we have the following corollary.
Corollary 17. Let X, Y be
$T_0$
spaces and
$x\in X$
. Then,
-
(1) The function
$E^{K}_x:[X\rightarrow Y]_{K}\rightarrow Y$
,
$f\mapsto f(x)$
is continuous. -
(2) The function
$E^{I}_x:[X\rightarrow Y]_{I}\rightarrow Y$
,
$f\mapsto f(x)$
is continuous.
In the following, for topological spaces X, Y and
$y\in Y$
,
$c_y$
denotes the constant function from X to Y with value y, i.e.,
$c_{y}(x)=y$
for all
$x\in X$
.
Lemma 18. (Liu et al. Reference Liu, Li and Ho2021, Lemma 3.2) Let X and Y be
$T_0$
spaces. Then, the function
$$\! \psi^{I}:Y\rightarrow[X\rightarrow Y]_{I}, y\! \mapsto \! c_{y},$$
is continuous.
Corollary 19. Let X, Y be T
0 spaces and
$\mathcal T$
a topology on
$\mathbb{C}(X, Y)$
which is finer than the upper topology and coarser than the Isbell topology. Then, the function
$\psi^{\mathcal T}: Y\rightarrow (\mathbb{C}(X, Y), \mathcal T)$
,
$y\mapsto c_{y}$
is continuous.
In particular, by Lemmas 15 and 18, we get the following corollary.
Corollary 20. Let X and Y be
$T_0$
spaces. Then
-
(1) The function
$\psi^{K}:Y\rightarrow[X\rightarrow Y]_{K}$
,
$y\mapsto c_{y}$
is continuous. -
(2) The function
$\psi^{P}:Y\rightarrow[X\rightarrow Y]_{P}$
,
$y\mapsto c_{y}$
is continuous.
Remark 21. The similar results of Lemma 18 and Corollary 20 for the Scott topology does not hold in general. Indeed, let
$X=1=\{0\}$
be the topological space with single point (clearly,
$\mathcal O (X)=\mathcal O(1)=\{\emptyset, 1\}$
and Y the space
$(P, \upsilon (P))$
, where P is a poset for which the Scott topology is strictly finer than the upper topology on P (e.g., P is a countably infinite set with the discrete order). Then,
$\mathbb{C}(X, Y)=\{c_y : y\in Y\}$
and
$c_y\mapsto y : [X\rightarrow Y]_{\Sigma}\rightarrow \Sigma Y=\Sigma P$
is a homeomorphism. Since
$\upsilon (P)\subseteq \sigma (P)$
, the function
$\psi^{\Sigma}: Y\rightarrow[X\rightarrow Y]_{\Sigma}$
,
$y\mapsto c_{y}$
is not continuous.
Lemma 22. Let X, Y be
$T_0$
spaces and
$\mathcal T$
a topology on T0 which is finer than the upper topology and coarser than the Isbell topology. Then, Y is a retract of
$(\mathbb{C}(X, Y), \mathcal T)$
. In particular, Y is a retract of
$[X\rightarrow Y]_P$
(resp.,
$[X\rightarrow Y]_K$
,
$[X\rightarrow Y]_I$
).
Proof. Select an
$x\in X$
. Then by Lemma 16 and Corollary 19,
$E^{\mathcal T}_x: (\mathbb{C}(X, Y), \mathcal T) \rightarrow Y$
and
$\psi^{\mathcal T}: Y\rightarrow (\mathbb{C}(X, Y), \mathcal T)$
are continuous. Clearly,
$E^{\mathcal T}_x \circ \psi^{\mathcal T}=id_{Y}$
. Thus, Y is a retract of
$(\mathbb{C}(X, Y), \mathcal T)$
.
Lemma 23. Let X, Y be
$T_0$
spaces and
$\emptyset\neq\mathcal {F}\subseteq \mathbb{C}(X, Y)$
. For any mapping
$g : X \rightarrow Y$
, the following two conditions are equivalent:
-
(1) For each
$x\in X$
,
$\overline{\{f(x) : f\in \mathcal{F}\}}=\overline{\{g(x)\}}$
. -
(2) For each
$V\in \mathcal O(Y)$
,
$g^{-1}(V)=\bigcup\limits_{f\in \mathcal F}f^{-1}(V)$
.
Therefore, when condition (1) is satisfied, we have that
$g\in \mathbb{C}(X, Y)$
,
$g(x)=\bigvee_{f\in \mathcal{F}} f(x)$
for each
$x\in X$
and
$g=\bigvee_{\mathbb{C}(X, Y)}\mathcal{F}$
.
Proof. (1)
$\Rightarrow$
(2): Let
$V\in \mathcal O(Y)$
. If
$x\in g^{-1}(V)$
or, equivalently,
$g(x)\in V$
, then by (1),
$\{f(x) : f\in \mathcal{F}\}\cap V\neq \emptyset$
and hence
$f_V(x)\in V$
for some
$f_V\in \mathcal{F}$
. So
$x\in f_V^{-1}(V)\subseteq \bigcup\limits_{f\in \mathcal F}f^{-1}(V)$
. Thus,
$g^{-1}(V)\subseteq\bigcup\limits_{f\in \mathcal F}f^{-1}(V)$
. Conversely, if
$x\in \bigcup\limits_{f\in \mathcal F}f^{-1}(V)$
, then
$f_x(x)\in V$
for some
$f_x\in \mathcal {F}$
. By (1),
$f_x(x)\in \overline{\{g(x)\}}$
; whence
$g(x)\in V$
, that is,
$x\in g^{-1}(V)$
. Therefore,
$\bigcup\limits_{f\in \mathcal F}f^{-1}(V)\subseteq g^{-1}(V)$
. Thus,
$g^{-1}(V)=\bigcup\limits_{f\in \mathcal F}f^{-1}(V)$
.
(2)
$\Rightarrow$
(1): For each
$x\in X$
and
$V\in \mathcal O(Y)$
, by (2) we have that
$\{g(x)\}\cap V\neq\emptyset$
iff
$x\in g^{-1}(V)$
iff
$x\in f_x^{-1}(V)$
for some
$f_x\in \mathcal F$
iff
$\{f(x) : f\in \mathcal{F}\}\cap V\neq\emptyset$
. So
$\overline{\{f(x) : f\in \mathcal{F}\}}=\overline{\{g(x)\}}$
.
Now suppose that condition (1) is satisfied (and hence condition (2) holds). Then, by (2)
$g : X \rightarrow Y$
is continuous, and for each
$x\in X$
, as
$f(x)\in\overline{\{g(x)\}}$
for each
$f\in \mathcal F$
, g(x) is an upper bound of
$\{f(x) : f\in \mathcal{F}\}$
. If
$y\in Y$
is also an upper bound of
$\{f(x) : f\in \mathcal{F}\}$
, then
$g(x)\in \overline{\{f(x) : f\in \mathcal{F}\}}\subseteq \overline{\{y\}}$
, whence
$g(x)\leq_Y y$
. So
$g(x)=\bigvee_{f\in \mathcal{F}} f(x)$
. Clearly, g is an upper bound of
$\mathcal{F}$
. If
$h\in \mathbb{C}(X, Y)$
is another upper bound of
$\mathcal{F}$
, then for each
$x\in X$
,
$\{f(x) : f\in\mathcal F\}\subseteq \overline{\{h(x)\}}$
; whence
$g(x)\in \overline {\{f(x) : f\in\mathcal F\}}\subseteq \overline{\{h(x)\}}$
, that is,
$g(x)\leq h(x)$
. Hence,
$g\leq h$
. So
$g=\bigvee_{\mathbb{C}(X, Y)}\mathcal{F}$
.
By Proposition 4 and Lemma 23, we get the following corollary.
Corollary 24. (Gierz et al. Reference Gierz, Hofmann, Keimel, Lawson, Mislove and Scott2003, Lemma II-3.14 and Proposition II-3.15) Let X be a
$T_0$
space and Y a d-space. Then,
-
(1) If
$(f_d)_{d\in D}$
is a net of continuous functions
$f_d : X\rightarrow Y$
such that
$(f_d(x))_{d\in D}$
is a directed net of Y (with the specialization order) for each
$x\in X$
, then the pointwise sup
$f : X\rightarrow Y$
of the net
$f_d$
is continuous and f is the least upper bound of
$(f_d)_{d\in D}$
in
$\mathbb{C}(X, Y)$
. -
(2) The subset
$\mathbb{C}(X, Y)$
of
$(\Omega Y)^X$
is closed under directed sups and hence
$\mathbb{C}(X, Y)$
is a dcpo.
Corollary 25. Let X be a
$T_0$
space and Y a d-space. Then, the function
$\psi^{\sigma}:\Sigma Y\rightarrow[X\rightarrow Y]_{\Sigma}$
,
$y\mapsto c_{y}$
is continuous.
Proof. For any
$D\in \mathcal D(Y)$
, by Corollary 24, we have
$\psi^{\sigma}(\vee_Y D)=c_{\vee_Y D}=\bigvee_{\mathbb{C}(X, Y)}\psi^{\sigma}(D)$
. Hence by Lemma 1,
$\psi^{\sigma}:\Sigma Y\rightarrow[X\rightarrow Y]_{\Sigma}$
is continuous. 
Motivated by Lemma 23, we give the following definition.
Definition 26. Let
${\rm H} : \mathbf{Top}_0 \rightarrow \mathbf{Set}$
be an R-subset system and X, Y
$T_{0}$
spaces. An order compatible topology
$\mathcal T$
on
$\mathbb{C}(X, Y)$
is said to have property S with respect to H if it satisfies the following condition:
(S) For any
$\mathcal F\in {\rm H}((\mathbb{C}(X, Y), \mathcal T))$
and
$g\in \mathbb{C}(X, Y)$
, if
$g^{-1}(V)=\bigcup_{f\in\mathcal F}f^{-1}(V)$
for each
$V\in \mathcal O(Y)$
, then
$g\in {\rm cl}_{\mathcal T}\mathcal F$
.
Remark 27. By the order compatibility of
$\mathcal T$
and Lemma 23,
$\mathcal T$
has property S with respect to H iff for any
$\mathcal F\in {\rm H}((\mathbb{C}(X, Y), \mathcal T))$
and
$g\in \mathbb{C}(X, Y)$
with
$g^{-1}(V)=\bigcup_{f\in\mathcal F}f^{-1}(V)$
for all
$V\in \mathcal O(Y)$
, we have
${\rm cl}_{\mathcal T}\mathcal F={\rm cl}_{\mathcal T}\{g\}$
.
Lemma 28. Let X be a
$T_0$
space and Y a d-space and
$x\in X$
. Then,
$\mathbb{C}(X, Y)$
is a dcpo and the following two functions are continuous:
-
(1)
$E^{\sigma}_x : [X\rightarrow Y]_{\Sigma} \rightarrow \Sigma Y, f\mapsto f(x)$
. -
(2)
$E^{\Sigma}_x: [X\rightarrow Y]_{\Sigma} \rightarrow Y, f\mapsto f(x)$
.
Proof. By Corollary 24,
$\mathbb{C}(X, Y)$
is a dcpo. Clearly,
$E^{\sigma}_x$
is monotone. For any
$\{f_d : d\in D\}\in \mathcal D({\mathbb{C} (X, Y)})$
, by Corollary 24, we have
$E^{\sigma}_x(\bigvee_{\mathbb{C}(X, Y)}\{f_d : d\in D\})=\bigvee_{d\in D}f_d(x)=\bigvee_Y \{E^{\sigma}_x(f_d) : d\in D\}$
. Hence by Lemma 1,
$E^{\sigma}_x : [X\rightarrow Y]_{\Sigma} \rightarrow \Sigma Y$
is continuous; whence by
$\mathcal O(Y)\subseteq \sigma (Y)$
,
$E^{\Sigma}_x: [X\rightarrow Y]_{\Sigma} \rightarrow Y$
is continuous.
Corollary 29. Let X be a
$T_0$
space and Y a d-space. Then,
-
(1)
$\Sigma Y$
is a retract of
$[X\rightarrow Y]_{\Sigma}$
. -
(2) If
$\psi^{\Sigma} : Y\rightarrow[X\rightarrow Y]_{\Sigma}$
,
$y\mapsto c_{y}$
is continuous, then Y is a retract of
$[X\rightarrow Y]_{\Sigma}$
.
Proof. (1): Select an
$x\in X$
. Then by Lemmas 16 and Corollary 25, both the function
$E^{\sigma}_x : [X\rightarrow Y]_{\Sigma} \rightarrow \Sigma Y$
and the function
$\psi^{\sigma} : \Sigma Y\rightarrow[X\rightarrow Y]_{\Sigma}$
are continuous. Clearly,
$E^{\sigma}_x \circ \psi^{\sigma}=id_{\Sigma Y}$
. Thus,
$\Sigma Y$
is a retract of
$[X\rightarrow Y]_{\Sigma}$
.
(2): Suppose that
$\psi^{\Sigma} : Y\rightarrow[X\rightarrow Y]_{\Sigma}$
is continuous. By Lemma 28,
$E^{\Sigma}_x: [X\rightarrow Y]_{\Sigma} \rightarrow Y$
is continuous. Clearly,
$E^{\Sigma}_x \circ \psi^{\Sigma}=id_{Y}$
. So Y is a retract of
$[X\rightarrow Y]_{\Sigma}$
.
Lemma 30. Let X, Y be
$T_0$
spaces and
$V\in \mathcal O(Y)$
. Then, the function
$$\Theta^I_V:[X\rightarrow Y]_{I}\rightarrow \Sigma \mathcal{O}(X), f\mapsto f^{-1}(V),$$
is continuous.
Proof. For each
$\mathcal H\in \sigma(\mathcal{O}(X))$
, we have
$(\Theta^I_V)^{-1}(\mathcal H)=\{f\in \mathbb{C}(X, Y) : f^{-1}(V)\in \mathcal H\}= N(\mathcal H\leftarrow V)$
, which is open in
$[X\rightarrow Y]_{I}$
. Thus,
$\Theta^I_V : [X\rightarrow Y]_{I}\rightarrow \Sigma \mathcal{O}(X)$
is continuous.
Lemma 31. Let X be a
$T_0$
space and Y a d-space. Then for any
$V\in \mathcal O(Y)$
, the function
$$\Theta^\Sigma_V:[X\rightarrow Y]_{\Sigma}\rightarrow \Sigma \mathcal{O}(X), f\mapsto f^{-1}(V),$$
is continuous.
Proof. By Corollary 24,
$\mathbb{C}(X, Y)$
is a dcpo and the directed sups in
$\mathbb{C}(X, Y)$
is the pointwise sups. So for any directed family
$\{f_d : d\in D\}\subseteq \mathbb{C}(X, Y)$
, by
$\mathcal O(Y)\subseteq \sigma (Y)$
, we have that
$\Theta^\Sigma_V (\bigvee_{\mathbb{C}(X, Y)}\{f_d : d\in D\})=(\bigvee_{\mathbb{C}(X, Y)}\{f_d : d\in D\})^{-1}(V)=\bigcup_{d\in D}(f_d)^{-1}(V)=\bigcup_{d\in D}\Theta^\Sigma_V(f_d)$
. Hence by Lemma 1,
$\Theta^\Sigma_V:[X\rightarrow Y]_{\Sigma}\rightarrow \Sigma \mathcal{O}(X)$
is continuous.
Lemma 32. Let X be a
$T_0$
space and Y a d-space having at least two points. Suppose that Y has a least element
$\bot_Y$
. Select any
$y\in Y$
with
$y\neq \bot_Y$
. Then, the function
$$\chi^y : \Sigma \mathcal{O}(X)\rightarrow[X\rightarrow Y]_{\Sigma}, U\mapsto \chi_U^y,$$
is continuous, where
$\chi_U^y : X \rightarrow Y$
is defined by
$$\chi_U^y(x)=\begin{cases} y,& x\in U,\\ \bot_Y,& x\not\in U. \end{cases}$$
Proof. For each
$U\in \mathcal O(X)$
and
$W\in \mathcal O(Y)$
, we clearly have
$$(\chi_U^y)^{-1}(W)=\begin{cases} X,& W=Y,\\ \emptyset,& W\neq Y, y\not\in U,\\ U, & W\neq Y, y\in U. \end{cases}$$
So
$\chi_U^y : X \rightarrow Y$
is continuous. For any directed family
$\{U_d : d\in D\}\subseteq \mathcal O(X)$
, let
$U=\bigcup_{d\in D}U_d$
. It is straightforward to verify that
$\chi^y (\bigcup_{d\in D}U_d)=\chi_U^y=\bigvee_{\mathbb{C}(X, Y)}\{\chi_{U_d}^y : d\in D\}$
. Hence by Lemma 1,
$\chi^y : \Sigma \mathcal{O}(X)\rightarrow[X\rightarrow Y]_{\Sigma}$
is continuous.
A
$T_0$
space X is said to be a non-trivial space if it has at least two points.
Definition 33. A T
0-space Y is called a pointed d-space if it is a non-trivial d-space and has a least element
$\bot_Y$
(with respect to the specialization order).
The simplest pointed d-space is the Sierpiński space
$\Sigma 2$
.
Corollary 34. Let X be a
$T_0$
space and Y a pointed d-space. Then,
$\Sigma \mathcal O(X)$
is a retract of
$[X\rightarrow Y]_\Sigma$
.
Proof. As Y is a pointed d-space,
$V=Y\setminus \{\bot_Y\}=Y\setminus {\rm cl}_Y\{\bot_Y\}$
is a non-empty open set of Y, where
$\bot_Y$
is the least element of Y. Select a
$y\in V$
. Then by Lemmas 31 and 32,
$\Theta^\Sigma_V:[X\rightarrow Y]_{\Sigma}\rightarrow \Sigma \mathcal{O}(X)$
and
$\chi^y : \Sigma \mathcal{O}(X)\rightarrow[X\rightarrow Y]_{\Sigma}$
are continuous. For each
$U\in \mathcal O(X)$
, we have
$\Theta^\Sigma_V(\chi^y(U))=\Theta^\Sigma_V(\chi_U^y)=(\chi_U^y)^{-1}(V)=U$
. So
$\Theta^\Sigma_V\circ \chi^y=id_{\Sigma \mathcal{O}(X)}$
. Thus,
$\Sigma \mathcal{O}(X)$
is a retract of
$[X\rightarrow Y]_\Sigma$
.
5. Function Spaces Equipped with Isbell Topology
Lemma 35. Let
${\rm H} : \mathbf{Top}_0 \rightarrow \mathbf{Set}$
be an R-subset system for which
$\mathcal S\leq {\rm H}\leq \mathsf{WD}$
and X, Y two
$T_{0}$
spaces. Then, the Isbell topology on
$\mathbb{C}(X, Y)$
has property S with respect to H.
Proof. Let
$\mathcal F\in {\rm H}([X\rightarrow Y]_{I})$
and
$g\in \mathbb{C}(X, Y)$
satisfying
$g^{-1}(V)=\bigcup_{f\in \mathcal F}f^{-1}(V)$
for each
$V\in \mathcal O(Y)$
. For any
$V\in \mathcal O(Y)$
, by Lemma 30, we have
$\Theta^{I}_V(\mathcal F)=\{f^{-1}(V) : f\in \mathcal F\}\in {\rm H}(\Sigma \mathcal O(X))$
. By Proposition 5,
$\Sigma \mathcal O(X)$
is well-filtered (i.e.,
$\mathsf{WD}$
-sober) and hence H-sober since
$\mathcal S\leq {\rm H}\leq \mathsf{WD}$
. So there is a unique
$W_V\in \mathcal O(X)$
such that
${\rm cl}_{\sigma (\mathcal O(X))}\{f^{-1}(V) : f\in \mathcal F\}={\rm cl}_{\sigma (\mathcal O(X))}\{W_V\}$
, and consequently,
$g^{-1}(V)=\bigcup_{f\in \mathcal F}f^{-1}(V)=\bigcup {\rm cl}_{\sigma (\mathcal O(X))}\{f^{-1}(V) : f\in \mathcal F\}=\bigvee {\rm cl}_{\sigma (\mathcal O(X))}\{W_V\}=W_V$
. Hence,
${\rm cl}_{\sigma (\mathcal O(X))}\{f^{-1}(V) : f\in \mathcal F\}={\rm cl}_{\sigma (\mathcal O(X))}\{g^{-1}(V)\}$
.
Now we verify that
$g\in {\rm cl}_{[X\rightarrow Y]_{I}}\mathcal F$
. For any subbasic open set
$N(\mathcal H\leftarrow W)$
in
$[X\rightarrow Y]_{I}$
with
$g\in N(\mathcal H\leftarrow W)$
(where
$\mathcal H\in \sigma(\mathcal{O}(X))$
and
$W\in \mathcal O(Y)$
), we have
$g^{-1}(W)\in \mathcal H$
; whence by
$g^{-1}(W)\in {\rm cl}_{\sigma (\mathcal O(X))}\{f^{-1}(W) : f\in \mathcal F\}$
, there is
$f_W\in \mathcal F$
such that
$f_W^{-1}(W)\in \mathcal H$
. Thus,
$\mathcal F\bigcap N(\mathcal H\leftarrow W)\neq \emptyset$
. As
$\mathcal {F}\in H([X\rightarrow Y]_{I})\subseteq {\mathsf{Irr}} ([X\rightarrow Y]_{I})$
, all basic open sets of g in
$[X\rightarrow Y]_{I}$
must meet
$\mathcal {F}$
. So
$g\in {\rm cl}_{[X\rightarrow Y]_{I}}\mathcal {F}$
, proving that the Isbell topology on
$\mathbb{C}(X, Y)$
has property S with respect to H.
However, for
$T_0$
spaces X and Y, the Isbell topology on
$\mathbb{C}(X, Y)$
does not have property S with respect to
$\mathcal R$
in general (see Proposition 39 below).
Theorem 36. Let
${\rm H} : \mathbf{Top}_0 \rightarrow \mathbf{Set}$
be an R-subset system and X a
$T_{0}$
space. Suppose that
$\mathcal T$
is a topology on
$\mathbb{C}(X, Y)$
which is finer than the pointwise convergence topology. Consider the following two conditions:
-
(1)
$(\mathbb{C}(X, Y), \mathcal T)$
is H-sober.
-
(2)
$\mathcal T$
has property S with respect to H.
Then (1)
$\Rightarrow$
(2). Moreover, if Y is H-sober, then the two conditions are equivalent.
Proof. (1)
$\Rightarrow$
(2): Let
$\mathcal F\in {\rm H}((\mathbb{C}(X, Y), \mathcal T))$
and
$g\in \mathbb{C}(X, Y)$
satisfying
$g^{-1}(V)=\bigcup_{f\in \mathcal F}f^{-1}(V)$
for each
$V\in \mathcal O(Y)$
. By the H-sobriety of
$(\mathbb{C}(X, Y), \mathcal T)$
, there is a unique
$g^{\prime}\in \mathbb{C}(X, Y)$
such that
${\rm cl}_{\mathcal T}\mathcal F={\rm cl}_{\mathcal T}\{g^{\prime}\}$
. Hence,
$g^{\prime}=\bigvee_{\mathbb{C}(X, Y)}\mathcal{F}=g$
by Lemma 23. So
${\rm cl}_{\mathcal T}\mathcal F={\rm cl}_{\mathcal T}\{g\}$
.
(2)
$\Rightarrow$
(1): Suppose that Y is H-sober. For any
$\mathcal F\in {\rm H}((\mathbb{C}(X, Y), \mathcal T))$
and
$x\in X$
, by Lemma 16, we have that
$E_x^{\mathcal T}(\mathcal F)=\{f(x) : f\in \mathcal F\}\in {\rm H}(Y)$
. As Y is H-sober, there is a unique point
$y_{x}\in Y$
such that
$\overline{\{f(x) : f\in \mathcal {\mathcal {F}}\}}=\overline{\{y_{x}\}}$
. Now we can define a function
$$g:X\rightarrow Y \ \ by \ \ g(x)=y_{x} \hbox{ for each } x\in X.$$
Then by Lemma 23,
$g : X\rightarrow Y$
is continuous and
$g^{-1}(V)=\bigcup_{f\in \mathcal F}f^{-1}(V)$
for each
$V\in \mathcal O(Y)$
. Since
$\mathcal T$
has property S with respect to H, we have
$g\in {\rm cl}_{\mathcal T}\mathcal F$
or, equivalently,
${\rm cl}_{\mathcal T}\mathcal F={\rm cl}_{\mathcal T}\{g\}$
. Thus,
$(\mathbb{C}(X, Y), \mathcal T)$
is H-sober.
Theorem 37. Let
${\rm H} : \mathbf{Top}_0 \rightarrow \mathbf{Set}$
be an R-subset system and X a
$T_{0}$
space. Suppose that
$(\mathcal T)$
is a topology on
$\mathbb{C}(X, Y)$
which is finer than the pointwise convergence topology and coarser than the Isbell topology. Consider the following two conditions:
-
(1)
$(\mathbb{C}(X, Y), \mathcal T)$
is H-sober. -
(2) Y is H-sober.
Then, (1)
$\Rightarrow$
(2). Moreover, if
$\mathcal T$
has property S with respect to H, then the two conditions are equivalent.
Proof. (1)
$\Rightarrow$
(2): By Lemmas 12 and 22.
(2)
$\Rightarrow$
(1): Suppose that
$\mathcal T$
has property S with respect to H and Y is H-sober. Then by Theorem 36,
$(\mathbb{C}(X, Y), \mathcal T)$
is H-sober.
Furthermore, we have the following result.
Theorem 38. Let
${\rm H} : \mathbf{Top}_0 \rightarrow \mathbf{Set}$
be an R-subset system and X a
$T_{0}$
space. Then the following three conditions are equivalent:
-
(1)
$\Sigma \mathcal O(X)$
is H-sober.
-
(2) For every H-sober space Y, the function space
$[X\rightarrow Y]_{I}$
is H-sober. -
(3) For every H-sober space Y, the Isbell topology on
$\mathbb{C}(X, Y)$
has property S with respect to H.
Proof. (1)
$\Rightarrow$
(2): Suppose that
$\Sigma \mathcal O(X)$
and Y are H-sober. We will show that
$[X\rightarrow Y]_{I}$
is H-sober. For any
$\mathcal F\in {\rm H}([X\rightarrow Y]_{I})$
and
$x\in X$
, by Corollary 17,
$E^{I}_x(\mathcal F)=\{f(x) : f\in \mathcal F\}\in {\rm H}(Y)$
; whence by the H-sobriety of Y, there is a unique
$y_x\in Y$
such that
$\overline{\{f(x) : f\in \mathcal {\mathcal {F}}\}}=\overline{\{y_{x}\}}$
. Now, we can define a function
$g:X\rightarrow Y$
by
$g(x)=y_{x}$
for each
$x\in X$
. Then by Lemma 23,
$g\in\mathbb{C}(X, Y)$
and
$g^{-1}(V)=\bigcup_{f\in \mathcal F}f^{-1}(V)$
for each
$V\in \mathcal O(Y)$
.
Now we prove
${\rm cl}_{[X\rightarrow Y]_{I}}\{g\}={\rm cl}_{[X\rightarrow Y]_{I}}\mathcal {F}$
. Clearly,
$f\leq g$
for each
$f\in \mathcal F$
; whence
${\rm cl}_{[X\rightarrow Y]_{I}}\mathcal {F}\subseteq {\rm cl}_{[X\rightarrow Y]_{I}}\{g\}$
. Then, we show
$g\in {\rm cl}_{[X\rightarrow Y]_{I}}\mathcal {F}$
. First, for
$V\in \mathcal O(Y)$
, by Lemma 30,
$\Theta^I_V(\mathcal F)=\{f^{-1}(V) : f\in \mathcal F\}\in {\rm H}(\Sigma \mathcal O(X))$
. As
$\Sigma \mathcal O(X)$
is H-sober, there is a unique
$W_V\in \mathcal O(X)$
such that
${\rm cl}_{\sigma (\mathcal O(X))}\{f^{-1}(V) : f\in \mathcal F\}={\rm cl}_{\sigma (\mathcal O(X))}\{W_V\}$
, and consequently,
$g^{-1}(V)=\bigcup_{f\in \mathcal F}f^{-1}(V)=\bigcup {\rm cl}_{\sigma (\mathcal O(X))}\{f^{-1}(V) : f\in \mathcal F\}=\bigvee {\rm cl}_{\sigma (\mathcal O(X))}\{W_V\}=W_V$
. So
${\rm cl}_{\sigma (\mathcal O(X))}\{f^{-1}(V) : f\in \mathcal F\}={\rm cl}_{\sigma (\mathcal O(X))}\{g^{-1}(V)\}$
. Then, we verify that
$g\in {\rm cl}_{[X\rightarrow Y]_{I}}\mathcal F$
. For any subbasic open set
$N(\mathcal H\leftarrow W)$
in
$[X\rightarrow Y]_{I}$
with
$g\in N(\mathcal H\leftarrow W)$
(where
$\mathcal H\in \sigma(\mathcal{O}(X))$
and
$W\in \mathcal O(Y)$
), we have
$g^{-1}(W)\in \mathcal H$
; whence by
$g^{-1}(W)\in {\rm cl}_{\sigma (\mathcal O(X))}\{f^{-1}(W) : f\in \mathcal F\}$
, there is
$f_W\in \mathcal F$
such that
$f_W^{-1}(W)\in \mathcal H$
. Thus,
$\mathcal F\bigcap N(\mathcal H\leftarrow W)\neq \emptyset$
. As
$\mathcal {F}\in H([X\rightarrow Y]_{I})\subseteq {\mathsf{Irr}} ([X\rightarrow Y]_{I})$
, all basic open sets of g in
$[X\rightarrow Y]_{I}$
must meet
$\mathcal {F}$
. So
$g\in {\rm cl}_{[X\rightarrow Y]_{I}}\mathcal {F}$
. Thus,
${\rm cl}_{[X\rightarrow Y]_{I}}\mathcal {F}={\rm cl}_{[X\rightarrow Y]_{I}}\{g\}$
, proving that
$[X\rightarrow Y]_{I}$
is H-sober.
(2)
$\Rightarrow$
(1): As the Sierpiński space
$\Sigma 2$
is sober (and hence H-sober), by (2)
$[X\rightarrow \Sigma 2]_{I}$
is H-sober. But
$[X\rightarrow \Sigma 2]_{I}$
is homeomorphic to
$\Sigma \mathcal O(X)$
, so
$\Sigma \mathcal O(X)$
is H-sober.
(2)
$\Leftrightarrow$
(3): By Theorem 36.
In Xu et al. (Reference Xu, Xi and Zhao2021), it was shown that there exist some
$T_0$
spaces X for which
$\Sigma \mathcal O(X)$
is non-sober. Using such spaces, we can give the following result.
Proposition 39. Let X be any
$T_0$
space for which
$\Sigma \mathcal O(X)$
is non-sober. Then,
-
(1)
$[X\rightarrow \Sigma 2]_I$
is not sober.
-
(2) The Isbell topology on
$\mathbb{C}(X, \Sigma 2)$
does not have property S with respect to
$\mathcal R$
.
Proof. (1): It is straightforward to verify that
$f\mapsto f^{-1}(1) : [X\rightarrow \Sigma 2]_I\rightarrow \Sigma \mathcal O(X)$
is a homeomorphism. As
$\Sigma \mathcal O(X)$
is non-sober,
$[X\rightarrow \Sigma 2]_I$
is not sober.
(2): Assume, on the contrary, that the Isbell topology on
$\mathbb{C}(X, \Sigma 2)$
has property S with respect to
$\mathcal R$
; then by the sobriety of
$\Sigma 2$
and Theorem 38,
$[X\rightarrow \Sigma 2]_I$
is sober, a contradiction. So the Isbell topology on
$\mathbb{C}(X, \Sigma 2)$
does not have property S with respect to
$\mathcal R$
.
By Lemmas 15, 35 and Theorem 37, we get the following three corollaries.
Corollary 40. Let X, Y be
$T_{0}$
spaces.
-
(1) If the function space
$[X\rightarrow Y]_{P}$
is H-sober, then Y is H-sober.
-
(2) If the function space
$[X\rightarrow Y]_{K}$
is H-sober, then Y is H-sober. -
(3) If the function space
$[X\rightarrow Y]_{I}$
is H-sober, then Y is H-sober.
Corollary 41. (Gierz et al. 2003, Lemma II-4.3) Let X, Y be
$T_{0}$
spaces. Then, the following two conditions are equivalent:
-
(1) Y is a d-space.
-
(2)
$[X\rightarrow Y]_{I}$
is a d-space, and hence, the Scott topology on
$\mathbb{C}(X, Y)$
is finer than the Isbell topology.
Corollary 42. Let X, Y be
$T_{0}$
spaces. Then, the following two conditions are equivalent:
-
(1) Y is well-filtered.
-
(2)
$[X\rightarrow Y]_{I}$
is well-filtered.
In Liu et al. (Reference Liu, Li and Ho2021, Corollary 3.12), it was proved that for a core-compact space X, if Y is well-filtered, then the function space
$\mathbb{C}(X, Y)$
equipped with the Isbell topology is a well-filtered space. For the well-filteredness, Corollary 42 improves this conclusion by removing the unnecessary condition that X is core-compact.
Applying Corollary 40 directly to the R-subset systems
$\mathcal{D}$
,
$\mathcal{R}$
and
$\mathsf{WD}$
(or
$\mathsf{RD}$
), we get the following corollary.
Corollary 43. Let X, Y be
$T_{0}$
spaces.
-
(1) If the function space
$[X\rightarrow Y]_{P}$
is a sober space (resp., d-space, well-filtered space), then Y is a sober space (resp., d-space, well-filtered space). -
(2) If the function space
$[X\rightarrow Y]_{K}$
is a sober space (resp., d-space, well-filtered space), then Y is a sober space (resp., d-space, well-filtered space). -
(3) If the function space
$[X\rightarrow Y]_{I}$
is a sober space, then Y is a sober space.
From Theorem 38 and Corollary 40, we deduce the following result.
Corollary 44. Let
$H:\mathbf{Top}_{0}\rightarrow \mathbf{Set}$
be an R-subset system and X a
$T_{0}$
space for which
$\Sigma \mathcal O(X)$
is H-sober. Then for any
$T_0$
space Y, the following two conditions are equivalent:
-
(1) Y is H-sober.
-
(2) The function space
$[X\rightarrow Y]_I$
is H-sober.
In particular, we have the following result.
Corollary 45. For a
$T_{0}$
space X, the following two conditions are equivalent:
-
(1)
$\Sigma \mathcal O(X)$
is sober.
-
(2) For every sober space Y, the function space
$[X\rightarrow Y]_I$
is sober.
6. Function Spaces Equipped with Pointwise Convergence Topology and Compact-Open Topology
In this section, we discuss the function spaces equipped with pointwise convergence topology and compact-open topology.
Proposition 46. Let
${\rm H} : \mathbf{Top}_0 \rightarrow \mathbf{Set}$
be an R-subset system and X, Y two
$T_{0}$
spaces. Then, the pointwise convergent topology on
$\mathcal T$
has property S with respect to H.
Proof. Let
$\mathcal F\in {\rm H}([X\rightarrow Y]_{P})$
and
$g\in \mathbb{C}(X, Y)$
satisfying
$g^{-1}(V)=\bigcup_{f\in \mathcal F}f^{-1}(V)$
for each
$V\in \mathcal O(Y)$
. Then by Lemma 23,
$\overline{\{g(x)\}}=\overline{\{f(x) : f\in \mathcal F\}}$
for each
$x\in X$
. For any subbasic open set S(x, W) in
$[X\rightarrow Y]_{P}$
with
$g\in S(x, W)$
(where
$W\in \mathcal O(Y)$
), we have
$g(x)\in W$
; whence by
$g(x)\in\overline{\{f(x) : f\in \mathcal F\}}$
, there is
$f_W\in \mathcal F$
such that
$f_W(x)\in W$
, that is,
$f_W\in S(x, W)$
. Thus,
$\mathcal F\bigcap S(x, W)\neq \emptyset$
. As
$\mathcal {F}\in H([X\rightarrow Y]_{P})\subseteq {\mathsf{Irr}} ([X\rightarrow Y]_{P})$
, all basic open sets of g in
$[X\rightarrow Y]_{P}$
must meet
$\mathcal {F}$
. So
$g\in {\rm cl}_{[X\rightarrow Y]_{P}}\mathcal {F}$
, proving that the pointwise convergent topology on
$\mathbb{C}(X, Y)$
has property S with respect to H.
Proposition 47. Let X, Y be
$T_{0}$
spaces. Then, the compact-open topology on
$\mathbb{C}(X, Y)$
has property S with respect to
$\mathcal D$
.
Proof. Let
$\{f_d : d\in D\}\in \mathcal D(\mathbb{C}(X, Y))$
and
$g\in \mathbb{C}(X, Y)$
satisfying
$g^{-1}(V)=\bigcup_{d\in D}f_d^{-1}(V)$
for each
$V\in \mathcal O(Y)$
. Then by Lemma 23,
$\overline{\{g(x)\}}=\overline{\{f_d(x) : d\in D\}}$
for each
$x\in X$
and hence
$g=\bigvee_{\mathbb{C}(X, Y)}\{f_d : d\in D\}$
. Now we show that
$g\in {\rm cl}_{[X\rightarrow Y]_K}\{f_d : d\in D\}$
. For each compact subset K of X and
$V\in \mathcal O(Y)$
with
$g\in N(K\rightarrow V)$
, we have
$K\subseteq g^{-1}(V)=\bigcup_{d\in D}f_d^{-1}(V)$
. As K is compact and
$\{f_d^{-1}(V) : d\in D\}$
is a directed family of open sets of X, there is
$d\in D$
such that
$K\subseteq f_d^{-1}(V)$
or, equivalently,
$f_d\in N(K\rightarrow V)$
. Hence
$\{f_d : d\in D\}\bigcap N(K\rightarrow V)\neq\emptyset$
. Since
$\{f_d : d\in D\}\in \mathcal D(\mathbb{C}(X, Y))$
, all basic open sets of g in
$[X\rightarrow Y]_{K}$
must meet
$\{f_d : d\in D\}$
. So
$g\in {\rm cl}_{[X\rightarrow Y]_{K}}\{f_d : d\in D\}$
, proving that the compact-open topology on
$\mathbb{C}(X, Y)$
has property S with respect to
$\mathcal D$
.
Question 48. For
$T_0$
spaces X, Y, whether the compact-open topology on
$\mathbb{C}(X, Y)$
has property S with respect to
$\mathsf{WD}$
or
$\mathsf{RD}$
?
Question 49. For
$T_0$
spaces X, Y, whether the compact-open topology on
$\mathbb{C}(X, Y)$
has property S with respect to
$\mathcal R$
?
From Theorem 37 and Proposition 46 (or Corollary 40) we deduce the following result.
Proposition 50. (Xu Reference Xu2021, Theorem 4.28) Let
$H:\mathbf{Top}_{0}\rightarrow \mathbf{Set}$
be an R-subset system and X, Y two
$T_{0}$
spaces. Then the following two conditions are equivalent:
-
(1) Y is H-sober.
-
(2) The function space
$[X\rightarrow Y]_P$
is H-sober.
Applying Corollary 40 and Proposition Proposition 50 directly to the R-subset systems
$\mathcal{D}$
,
$\mathcal{R}$
and
$\mathsf{WD}$
(or
$\mathsf{RD}$
), we get the following corollary.
Corollary 51. For
$T_0$
spaces X and Y, the following two conditions are equivalent:
-
(1) Y is a sober space (resp., d-space, well-filtered space).
-
(2) The function space
$[X\rightarrow Y]_P$
is a sober space (resp., d-space, well-filtered space).
The results for d-spaces and sober spaces in Corollary 51 were first shown in Ershov et al. (2020, Theorems 3 and 6) by a different method.
By Lemma 15, Theorem 37 and Proposition 47, we have the following result.
Corollary 52. For:
$T_0$
spaces X and Y, the following two conditions are equivalent
-
(1) Y is a d-space.
-
(2) The function space
$[X\rightarrow Y]_K$
is a d-space.
Remark 53. We can give a direct proof of Corollary 52.
Proof. Suppose that X is a
$T_0$
space and Y is a d-space. We show that the function space
$\mathbb{C}(X, Y)$
equipped with the compact-open topology is a d-space.
Let
$\mathcal {F}\in \mathcal{D}([X\rightarrow Y]_{K})$
. Since the specialization order on
$[X\rightarrow Y]_{K}$
is the usually pointwise order on
$\mathbb{C}(X, Y)$
, we have that for each
$x\in X$
,
$\{f(x): f\in \mathcal {F}\}\in \mathcal{D}(Y)$
. As Y is a d-space, there exists a unique point
$a_{x}\in Y$
such that
$\overline{\{f(x):f\in \mathcal {F}\}}=\overline{\{a_{x}\}}$
. Then, we can define a function
$g:X\rightarrow Y$
by
$g(x)=a_{x}$
for each
$x\in X$
. By Lemma 23, g is continuous. Clearly,
${\rm cl}_{[X\rightarrow Y]_{K}}\mathcal {F}\subseteq {\rm cl}_{[X\rightarrow Y]_{K}}\{g\}$
.
Now we show that
$g\in {\rm cl}_{[X\rightarrow Y]_{K}}\mathcal {F}$
. Let
$N(K\rightarrow V)$
be a subbasic open set in
$[X\rightarrow Y]_{K}$
with
$g\in N(K\rightarrow V)$
, where K is a compact subset of X and V is an open subset of Y. Then for any
$x\in K$
,
$g(x)\in V$
. As
$\overline{\{f(x):f\in \mathcal {F}\}}=\overline{\{g(x)\}}$
, there exists
$f\in \mathcal {F}$
with
$f(x)\in V$
, whence
$x\in f^{-1}(V)$
. Thus,
$K\subseteq \bigcup_{f\in \mathcal {F}}f^{-1}(V)$
. By the compactness of K, there exists
$\{f_{1}, f_{2},\cdots, f_{n}\}\subseteq \mathcal{F}$
such that
$K\subseteq \bigcup_{i=1}^{n}f_{i}^{-1}(V)$
. Since
$\mathcal {F}$
is directed in
$[X\rightarrow Y]_{K}$
, there is
$h\in \mathcal {F}$
such that
$f_{i}\leq h$
for all
$i\in \{1, 2, \cdots, n\}$
. Therefore,
$K\subseteq \bigcup_{i=1}^{n}f_{i}^{-1}(V)\subseteq h^{-1}(V)$
(note that open sets in Y are upper sets with the specialization order of Y). It follows that
$h\in N(K\rightarrow V)$
. So
$\mathcal {F}\cap N(K\rightarrow V)\neq \emptyset$
. Since
$\mathcal {F}\in \mathcal{D}([X\rightarrow Y]_{K})\subseteq {\mathsf{Irr}} ([X\rightarrow Y]_{K})$
and all basic open sets of g in
$[X\rightarrow Y]_{K}$
must meet
$\mathcal {F}$
, we get that
$g\in {\rm cl}_{[X\rightarrow Y]_{K}}\mathcal {F}$
, and hence,
${\rm cl}_{[X\rightarrow Y]_{K}}\mathcal {F}={\rm cl}_{[X\rightarrow Y]_{K}}\{g\}$
. Thus, the function space
$[X\rightarrow Y]_{K}$
is a d-space.
But we do not know whether the converse of Proposition 40 (2) holds for a general R-subset system H. So we pose the following question.
Question 54. For an R-subset system H, a
$T_{0}$
space X and an H-sober space Y, is the function space
$[X\rightarrow Y]_{K}$
H-sober?
Especially, we have the following two questions.
Question 55. For a
$T_0$
space X and a sober space Y, is the function space
$[X\rightarrow Y]_K$
sober?
Question 56. For a
$T_{0}$
space X and a well-filtered space Y, is the function space
$[X\rightarrow Y]_{K}$
well-filtered?
Note that when Y is the Sierpiński space
$\Sigma 2$
, the compact-open topology on
$[X\rightarrow <italic>Y</italic>]$
takes
$\{N(K\rightarrow 2)=\mathcal O(X)\}\bigcup\{N(K\rightarrow \{1\}) : K\in \mathsf{K}(X)\}$
as a base. So
$[X\rightarrow \Sigma 2]_K$
is homeomorphic to the space
$(\mathcal O(X), \phi (\mathcal O(X)))$
. Then as a special case of the Question 48 (resp., Question 49, Question 55), Question 56, we have the following ones.
Question 57. For a
$T_0$
space X, whether the compact-open topology on
$\mathbb{C}(X, \Sigma 2)$
has property S with respect to
$\mathsf{WD}$
or
$\mathsf{RD}$
?
Question 58. For a
$T_0$
space X, whether the compact-open topology on
$\mathbb{C}(X, \Sigma 2)$
has property S with respect to
$\mathcal R$
?
Question 59. For a
$T_0$
space X, is the function space
$[X\rightarrow \Sigma 2]_K$
well-filtered? Or equivalently, is
$(\mathcal O(X), \phi (\mathcal O(X)))$
well-filtered?
Question 60. For a
$T_0$
space X, is the function space
$[X\rightarrow \Sigma 2]_K$
sober? Or equivalently, is the space
$(\mathcal O(X), \phi (\mathcal O(X)))$
sober?
7. Function Spaces Equipped with Scott Topology
Finally, we investigate the Scott topology on
$\mathbb{C}(X, Y)$
. First, by Lemma 15, Theorem 36 and Corollary 41, we have the following result.
Theorem 61. Let
${\rm H} : \mathbf{Top}_0 \rightarrow \mathbf{Set}$
be an R-subset system, X a
$T_{0}$
space and Y a d-space. Consider the following two conditions:
-
(1)
$[X\rightarrow Y]_\Sigma$
is H-sober. -
(2) The Scott topology on
$\mathbb{C}(X, Y)$
has property S with respect to H.
Then (1)
$\Rightarrow$
(2). Moreover, if Y is H-sober, then the two conditions are equivalent.
Proposition 62. Let X, Y be
$T_{0}$
spaces. Then, the Scott topology on
$\mathbb{C}(X, Y)$
has property S with respect to
$\mathcal D$
.
Proof. Let
$\{f_d : d\in D\}\in \mathcal D(\mathbb{C}(X, Y))$
and
$g\in \mathbb{C}(X, Y)$
satisfying
$g^{-1}(V)=\bigcup_{d\in D}f_d^{-1}(V)$
for each
$V\in \mathcal O(Y)$
. Then by Lemma 23,
$\overline{\{g(x)\}}=\overline{\{f_d(x) : d\in D\}}$
for each
$x\in X$
and
$g=\bigvee_{\mathbb{C}(X, Y)}\{f_d : d\in D\}$
. Now we show that
$g\in {\rm cl}_{\sigma(\mathbb{C}(X, Y))}\{f_d : d\in D\}$
. For each
$\mathcal U\in\sigma(\mathbb{C}(X, Y))$
with
$g\in \mathcal U$
, we have
$\bigvee_{\mathbb{C}(X, Y)}\{f_d : d\in D\}\in \mathcal U$
; whence
$f_d\in \mathcal U$
for some
$d\in D$
. So
$g\in {\rm cl}_{\sigma(\mathbb{C}(X, Y))}\{f_d : d\in D\}$
. Thus, the Scott topology on
$\mathbb{C}(X, Y)$
has property S with respect to
$\mathcal D$
.
For
$T_0$
spaces X and Y, the Scott topology on
$\mathbb{C}(X, Y)$
does not have property S with respect to
$\mathcal R$
in general. Indeed, for a
$T_0$
space X, it is easy to see the Scott topology agrees with the Isbell topology on
$\mathbb{C}(X, \Sigma 2)$
, namely
$[X\rightarrow \Sigma 2]_\Sigma=[X\rightarrow \Sigma 2]_I$
. Hence by Proposition 39, we have the following result.
Proposition 63. Let X be any
$T_0$
space for which
$\Sigma \mathcal O(X)$
is non-sober. Then,
-
(1)
$[X\rightarrow \Sigma 2]_\Sigma$
is not sober. -
(2) the Scott topology on
$\mathbb{C}(X, \Sigma 2)$
does not have property S with respect to
$\mathcal R$
.
We can give a more direct proof of Proposition 63 (not using Proposition 39). Clearly,
$f\mapsto f^{-1}(1) : [X\rightarrow \Sigma 2]_\Sigma\rightarrow \Sigma \mathcal O(X)$
is a homeomorphism. As
$\Sigma \mathcal O(X)$
is non-sober,
$[X\rightarrow \Sigma 2]_\Sigma$
is not sober. If the Scott topology on
$\mathbb{C}(X, \Sigma 2)$
has property S with respect to
$\mathcal R$
, then by Lemma 15, Theorem 36 and Corollary 41,
$[X\rightarrow \Sigma 2]_{\Sigma}$
is sober, a contradiction. So the Scott topology on
$\mathbb{C}(X, \Sigma 2)$
does not have property S with respect to
$\mathcal R$
.
Question 64. For
$T_0$
spaces X and Y, whether the Scott topology on
$\mathbb{C}(X, Y)$
has property S with respect to
$\mathsf{WD}$
or
$\mathsf{RD}$
? Especially, for a
$T_0$
space X and a well-filtered space Y, whether the Scott topology on
$\mathbb{C}(X, Y)$
has property S with respect to
$\mathsf{WD}$
or
$\mathsf{RD}$
?
By Corollary 24, we directly get the following result.
Proposition 65. For a
$T_0$
space X and a d-space Y, the function space
$[X\rightarrow Y]_\Sigma $
is a d-space.
Proposition 63 shows that for sober spaces, the similar result to Proposition 65 does not hold in general. The following is another counterexample.
Example 66. Let
$X=1$
be the topological space with single point and L the complete lattice constructed by Isbell in Isbell (Reference Isbell1982). It is well-known that
$\Sigma L$
is non-sober. Let
$Y=(L, \upsilon (L))$
. Then by Zhao et al. (2015, Corollary 4.10) or Xu et al. (Reference Xu, Shen, Xi and Zhao2020, Proposition 2.9), Y is sober. Clearly,
$\mathbb{C}(X, Y)=\{c_y : y\in Y\}$
and
$c_y\mapsto y : [X\rightarrow Y]_{\Sigma}\rightarrow \Sigma \Omega Y=\Sigma L$
is a homeomorphism. So Y is sober but the function space
$[X\rightarrow Y]_\Sigma$
is non-sober.
It is still not known whether a similar result to Proposition 65 holds for well-filtered spaces. That is, we have the following question.
Question 67. For a
$T_{0}$
space X and a well-filtered space Y, is the function space
$[X\rightarrow Y]_\Sigma$
well-filtered? Or equivalently, whether the Scott topology on
$\mathbb{C}(X, Y)$
has property S with respect to
$\mathsf{WD}$
or
$\mathsf{RD}$
?
Let
$\mathbb{J}=\mathbb{N}\times (\mathbb{N}\cup \{\omega\})$
with ordering defined by
$(j, k)\leq (m, n)$
iff
$j = m$
and
$k \leq n$
, or
$n =\omega$
and
$k\leq m$
.
$\mathbb{J}$
is a well-known dcpo constructed by Johnstone in (1981) which is not sober in its Scott topology.
Proposition 68. There is no well-filtered topology on
$\mathbb{J}$
which has the given order as its specialization order, namely, for any topology
$\upsilon (\mathbb{J})\subseteq \tau \subseteq \sigma (\mathbb{J})$
,
$(\mathbb{J}, \tau)$
is not well-filtered.
Proof. The set
$\mathbb{J}_{\max}=\{(n, \infty) : n\in\mathbb N \}$
is the set of all maximal elements of
$\mathbb{J}$
and
$\mathsf{K}(\Sigma \!\!\mathbb{J})=(2^{\mathbb{J}_{\max}} \setminus \{\emptyset\})\bigcup \mathbf{Fin} \!\mathbb{J}$
(see Lu et al. Reference Lu and Li2017, Example 3.1). Since
$\upsilon (\mathbb{J})\subseteq \tau \subseteq \sigma (\mathbb{J})$
, we have that
$2^{\mathbb{J}_{\max}}\setminus \{\emptyset\}\subseteq \mathsf{K}((\mathbb{J}, \tau))$
. Let
$\mathcal K=\{\mathbb{J}_{\max}\setminus F : F\in (\mathbb{J}_{\max})^{(<\omega)}\}$
. Then,
$\mathcal K\subseteq \mathsf{K}((\mathbb{J}, \tau))$
is a filtered family and
$\bigcap\mathcal{K}=\bigcap_{F\in (\mathbb{J}_{\max})^{(<\omega)}} (\mathbb{J}_{\max}\setminus F)=\mathbb{J}_{\max}\setminus \bigcup (\mathbb{J}_{\max})^{(<\omega)})=\emptyset$
, but
$\mathbb{J}_{\max}\setminus F\neq\emptyset$
for all
$F\in (\mathbb{J}_{\max})^{(<\omega)}$
. Therefore,
$(\mathbb{J}, \tau)$
is not well-filtered.
Corollary 69. (Gierz et al. 2003, Exercise II-3.16 (V)) There is no sober topology on
$\mathbb{J}$
which has the given order as its specialization order.
Question 70. Is there a well-filtered space Y such that
$\Sigma Y$
(i.e.,
$\Sigma \Omega Y$
) is not well-filtered? Or equivalently, is there a dcpo P and a topology
$\upsilon (P)\subseteq \tau\subseteq \sigma (P)$
such that
$(P, \tau)$
is well-filtered but
$(P, \sigma (P))$
is not well-filtered? In particular, is there a dcpo P such that
$(P, \upsilon (P))$
is well-filtered but
$(P, \sigma (P))$
is not well-filtered?
If the answer of Question 70 is “Yes,” then the answer of Question 67 is “No”! Indeed, suppose that Y is a well-filtered space for which the Scott space
$\Sigma Y$
is not well-filtered and
$X=1$
(the topological space with single point). Then, the function space
$\mathbb{C}(X, Y)$
equipped with the Scott topology is not well-filtered since
$[X\rightarrow Y]_{\Sigma}$
and
$\Sigma Y$
are homeomorphic (see Example 66).
Conversely, for
$\mathcal D\leq{\rm H}\leq\mathcal R$
(in particular,
${\rm H}=\mathcal D, \mathcal R, \mathsf{WD}$
or
$\mathsf {RD}$
), the following example shows that the H-sobriety of
$[X\rightarrow Y]_\Sigma$
does not imply the H-sobriety of Y in general.
Example 71. Let X be the topological space with single point and [0, 1] the unit closed interval with the usual order of reals. Then,
$\sigma ([0, 1])\neq \gamma ([0, 1])$
(note that
$\{1\}\in \gamma ([0, 1])$
but
$\{1\}\not\in \sigma ([0, 1])$
). Clearly, the Alexandroff space
$\Gamma [0, 1]$
is not a d-space (since
$\gamma ([0, 1])\not\subseteq\sigma ([0, 1])$
) and the Scott space
$\Sigma [0, 1]$
is sober; whence
$\Sigma [0, 1]$
is H-sober and
$\Gamma [0, 1]$
is not H-sober by
$\mathcal D\leq{\rm H}\leq\mathcal R$
. As
$[X\rightarrow \Gamma [0, 1]]_{\Sigma}$
is homeomorphic to
$\Sigma \Omega (\Gamma [0, 1])=\Sigma [0, 1]$
(see Example 66), the function space
$[X\rightarrow \Gamma [0, 1]]_{\Sigma}$
is H-sober but
$\Gamma [0, 1]$
is not H-sober.
Finally, by Lemma 12, Corollaries 29 and 34, we get the following two results.
Proposition 72. Let
${\rm H} : \mathbf{Top}_0 \rightarrow \mathbf{Set}$
be an R-subset system, X a
$T_{0}$
space and Y a d-space. If
$[X\rightarrow Y]_{\Sigma}$
is H-sober, then
$\Sigma Y$
is H-sober. In particular, for a dcpo P, if
$[X\rightarrow \Sigma P]_{\Sigma}$
is H-sober, then
$\Sigma P$
is H-sober.
Proposition 73. Let
${\rm H} : \mathbf{Top}_0 \rightarrow \mathbf{Set}$
be an R-subset system and X a
$T_{0}$
space and Y a pointed d-space. If
$[X\rightarrow Y]_{\Sigma}$
is H-sober, then
$\Sigma \mathcal O(X)$
is H-sober.
Corollary 74. Let X be a
$T_{0}$
space for which
$\Sigma \mathcal O(X)$
is non-sober. Then for any pointed d-space Y,
$[X\rightarrow Y]_{\Sigma}$
is not sober.
Acknowledgements
The authors would like to thank the referee for the numerous and very helpful suggestions that have improved this paper substantially.
