2. Preliminaries

In this section, we briefly recall some fundamental concepts and basic results about ordered structures and $T_0$ -spaces that will be used in the paper. For further details, we refer the reader to Gierz et al. (Reference Gierz, Hofmann, Keimel, Lawson, Mislove and Scott2003), Goubault-Larrecq (Reference Goubault-Larrecq2013).

For a poset $P$ and $A\subseteq P$ , let $\mathord{\downarrow }A=\{x\in P: x\leq a \mbox{ for some } a\in A\}$ 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 $\mathord{\uparrow }\{x\}$ . The set $A$ is called a lower set (resp., an upper set) if $A=\mathord{\downarrow }A$ (resp., $A=\mathord{\uparrow }A$ ). The family of all upper subsets of $P$ is denoted by $\mathbf{up}(P)$ . Let $P^{(\lt \omega )}=\{F\subseteq P : F\, \mbox{is a nonempty finite set}\}$ and $\mathbf{Fin} P=\{\uparrow F$ : $F\in P^{(\lt \omega )}\}$ . An upper set $B$ of $P$ is said to be finitely generated if there is $F\in P^{(\lt \omega )}$ such that $B=\mathord{\uparrow } F$ . For a nonempty subset $C$ of $P$ , define $\mathrm{max}(C)=\{c\in C : c\, \mbox{is a maximal element of}\, C\}$ and $\mathrm{min}(C)=\{c\in A : c\, \mbox{is a minimal element of}\, C\}$ .

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 $\mathbb{N}$ . 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 $\omega =|\mathbb{N}|$ .

A poset $P$ is called an inf semilattice (shortly a semilattice) if any two elements $a, b\in P$ have the greatest lower bound in $P$ , denoted by $a\wedge b$ . Dually, $P$ is a sup semilattice if any two elements $a, b\in P$ have the least upper bound in $P$ , denoted by $a\vee b$ . The poset $P$ is called sup complete, if every nonempty subset of $P$ has a sup (i.e., the least upper bound). In particular, a sup complete poset has a greatest element, the sup of $P$ . $P$ is called a complete lattice if every subset (including the empty set) has a sup and an inf. A totally ordered complete lattice is called a complete chain.

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, if for any $D\in \mathscr{D}(P)$ , $\bigvee D$ exists in $P$ . Clearly, a poset $Q$ is sup complete iff $Q$ is both a dcpo and a sup semilattice.

The category of all $T_0$ -spaces and continuous mappings is denoted by $\mathbf{Top}_0$ . For a $T_0$ -space $X$ , let $\mathscr{O}(X)$ (resp., $\mathscr{C}(X)$ ) be the set of all open subsets (resp., closed subsets) of $X$ . The closure of a subset $A$ in $X$ will be denoted by ${\textrm{cl}}_X A$ (or simply by ${\textrm{cl}} A$ if there is no ambiguity) or $\overline{A}$ , and the interior of $A$ will be denoted by ${\textrm{int}}_X A$ or simply by ${\textrm{int}} A$ . Let $\mathscr{D}_c(X)=\{\overline{D} : D\in \mathscr{D}(X)\}$ . We use $\leq _X$ to denote the specialization order of $X$ : $x\leq _X y$ iff $x\in \overline{\{y\}}$ . Clearly, all open sets (resp., closed sets) of $X$ are upper sets (resp., lower sets) of $X$ . 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). For a poset $P$ , a $T_0$ -topology $\tau$ on $P$ is said to be order-compatible if $\leq _{(P, \tau )}$ (shortly $\leq _\tau$ ) agrees with the original order on $P$ .

In what follows, 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. We will use $\Omega X$ or simply $X$ to denote the poset $(X, \leq _X)$ .

For a set $X$ and two topologies $\tau$ and $\nu$ on $X$ , the join $\tau \bigvee \nu$ is the topology generated by $\tau \bigcup \nu$ . It is the smallest topology on $X$ containing both $\tau$ and $\nu$ .

A subset $U$ of a poset $P$ is said to be Scott-open if (i) $U=\mathord{\uparrow }U$ , and (ii) for any directed subset $D$ with $\bigvee D$ existing, $\bigvee D\in U$ implies $D\cap U\neq \emptyset$ . All Scott-open subsets of $P$ form a topology, called the Scott topology on $P$ and denoted by $\sigma (P)$ . The space $\Sigma P=(P,\sigma (P))$ is called the Scott space of $P$ . A subset $C$ of $P$ is said to be Scott-compact if it is compact in $\Sigma P$ . For the chain $2 =\{0, 1\}$ (with the order $0\lt 1$ ), we have $\sigma (2)=\{\emptyset, \{1\}, \{0, 1\}\}$ . The space $\Sigma 2$ is well-known under the name of Sierpiński space.

The lower topology on $P$ , generated by $\{P\setminus \mathord{\uparrow } x : x\in P\}$ (as a subbase), is denoted by $\omega (P)$ . Dually, define the upper topology on $P$ (generated by $\{P\setminus \mathord{\downarrow } x : x\in P\}$ ) and denote it by $\upsilon (P)$ . The topology $\sigma (P)\bigvee \omega (P)$ is called the Lawson topology on $P$ and is denoted by $\lambda (P)$ . The collection of all upper sets of $P$ forms the (upper) Alexandroff topology $\alpha (P)$ . For a $T_0$ -topology $\tau$ on $P$ , it is easy to verify that $\tau$ is order-compatible iff $\upsilon (P) \subseteq \tau \subseteq \alpha (P)$ .

In the following, when a poset $P$ is considered as a $T_0$ -space, the topology on $P$ always refers to the Scott topology unless stated otherwise.

The following three results are well-known (see Gierz et al. Reference Gierz, Hofmann, Keimel, Lawson, Mislove and Scott2003, Proposition II-2.1, Theorem III-1.9, and Proposition VI-1.6). The first is a key feature of the Scott topology.

Rudin’s Lemma, given by Rudin (Reference Rudin1981), 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; Goubault-Larrecq Reference Goubault-Larrecq2013). On some occasions, we only need the following consequence of Rudin’s Lemma.

A $T_0$ -space $X$ is said to be hyper-sober if for any $F\in{\mathsf{Irr}} (X)$ , there is a unique $x\in F$ such that $F\subseteq{\textrm{cl}} \{x\}$ (cf. Zhao and Ho Reference Zhao and Ho2015).

For the following definition and related conceptions, please refer to Abramsky and Jung (Reference Abramsky and Jung1994), Gierz et al. (Reference Gierz, Hofmann, Keimel, Lawson, Mislove and Scott2003), Goubault-Larrecq (Reference Goubault-Larrecq2013).

It is well-known that every algebraic domain is a continuous domain and every continuous domain is a quasicontinuous domain but the converse implications do not hold in general (see Gierz et al. Reference Gierz, Lawson and Stralka1983, Reference Gierz, Hofmann, Keimel, Lawson, Mislove and Scott2003).

For the concepts in the following definition, please refer to Erné (Reference Erné2018), Gierz et al. (Reference Gierz, Hofmann, Keimel, Lawson, Mislove and Scott2003), Heckmann (Reference Heckmann1992), Heckmann and Keimel (Reference Heckmann and Keimel2013).

The following result is well-known (see Gierz et al. Reference Gierz, Lawson and Stralka1983, Reference Gierz, Hofmann, Keimel, Lawson, Mislove and Scott2003; Heckmann Reference Heckmann1992).

A $T_0$ -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. Gierz et al. Reference Gierz, Hofmann, Keimel, Lawson, Mislove and Scott2003; Wyler Reference Wyler1981).

The $d$ -space has the following useful property.

For a $T_0$ -space $X$ and a nonempty subset $A$ of $X$ , $A$ is irreducible if for any $\{F_1, F_2\}\subseteq \mathscr{C}(X)$ , $A \subseteq F_1\cup F_2$ 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 $X$ . Clearly, every subset of $X$ that is directed under $\leq _X$ is irreducible.

A topological space $X$ is called sober, if for any $A\in{\mathsf{Irr}}_c(X)$ , there is a unique point $x\in X$ such that $A=\overline{\{x\}}$ . It is straightforward to verify that every sober space is a $d$ -space (cf. Gierz et al. Reference Gierz, Hofmann, Keimel, Lawson, Mislove and Scott2003). For simplicity, if a dcpo has a sober (resp., non-sober) Scott topology, then we will call $P$ a sober (resp., non-sober) dcpo.

The following result is well-known.

For a $T_0$ -space $X$ , we shall use $\mathord{\mathsf{K}}(X)$ to denote the set of all nonempty compact saturated subsets of $X$ and endow it with the Smyth order $\sqsubseteq$ , that is, for $K_1,K_2\in \mathord{\mathsf{K}}(X)$ , $K_1\sqsubseteq K_2$ iff $K_2\subseteq K_1$ . Let $\mathscr{S}^u(X)=\{\mathord{\uparrow } x : x\in X\}$ and $\mathscr{S}^u_2(X)=\{\mathord{\uparrow } x\cap \mathord{\uparrow } y : x, y\in X\}$ . Obviously, $\mathscr{S}^u(X)\subseteq \mathscr{S}^u_2(X)$ . The space $X$ is called well-filtered if it is $T_0$ , and for any open set $U$ and filtered family $\mathscr{K}\subseteq \mathord{\mathsf{K}}(X)$ , $\bigcap \mathscr{K}{\subseteq } U$ implies $K{\subseteq } U$ for some $K{\in }\mathscr{K}$ . It is called coherent if the intersection of any two compact saturated sets of $X$ is compact.

The following result is well-known (see, e.g., Xi and Zhao Reference Xi and Zhao2017, Proposition 3 or Xu et al. Reference Xu, Shen, Xi and Zhao2020b, Lemma 2.6).

For a dcpo with well-filtered Scott topology, Jia et al. (2018) gave the following useful characterization of coherence of its Scott space.

It is well-known that every sober space is well-filtered and every well-filtered space is a $d$ -space. Kou (Reference Kou2001) gave the first example of a dcpo whose Scott space is well-filtered but non-sober. Another simpler dcpo whose Scott topology is well-filtered but not sober was presented in Zhao et al. (Reference Zhao, Xi and Chen2019). Jia (Reference Jia2018) constructed a countable infinite dcpo whose Scott topology is well-filtered but non-sober. It is worth noting that Johnstone (Reference Johnstone1981) constructed the first example of a dcpo whose Scott space is non-sober (indeed, it is not well-filtered) and Isbell (Reference Isbell1982) constructed a complete lattice whose Scott space is non-sober. Xi and Lawson (Reference Xi and Lawson2017) showed that every complete lattice is well-filtered in its Scott topology.

The following result is well-known (see Gierz et al. Reference Gierz, Hofmann, Keimel, Lawson, Mislove and Scott2003, Kou Reference Kou2001).

The above result was improved in Lawson et al. (Reference Lawson, Wu and Xi2020) and Xu et al. (Reference Xu, Shen, Xi and Zhao2020b) by two different methods.

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 ambiguity. 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 briefly by $P_S(X)$ , is called the Smyth power space or upper space of $X$ (cf. Heckmann Reference Heckmann1992; Schalk Reference Schalk1993). It is easy to verify that the specialization order of $P_S(X)$ is the Smyth order (i.e.,, $\leq _{P_S(X)}=\sqsupseteq$ ). The canonical mapping $\xi _X: X\longrightarrow P_S(X)$ , $x\mapsto \mathord{\uparrow } x$ , is a topological embedding (cf. Heckmann Reference Heckmann1992; Heckmann and Keimel Reference Heckmann and Keimel2013; Schalk Reference Schalk1993).

3. Property R and $\Omega ^*$ -Compactness

In this section, we consider $T_0$ -spaces and posets equipped with the Scott topology and equip them also with the lower topology denoted by $\omega$ . We study in particular property R and the property of $\Omega ^*$ -compactness, the equivalence between them, and their main properties.

The property R was first introduced in Xu (Reference Xu2016a, Definition 10.2.11) (see also Wen and Xu Reference Wen and Xu2018). Clearly, every $T_1$ -space has property R and the Sierpiński space $\Sigma 2$ has property R. We are particularly interested in the conditions under which a $T_0$ -space or a poset equipped with the Scott topology has property R.

We next recall the definition of $\Omega ^*$ -compactness from Lawson et al. (Reference Lawson, Wu and Xi2020).

And lastly, we introduce a new concept.

In what follows, we will be working with the lattice of closed sets of the lower topology on a $T_0$ -space $X$ resp. poset $P$ , which we denote by $\omega ^*(X)$ resp. $\omega ^*(P)$ . The following is a key theorem.

Proof. (1) $\Rightarrow$ (2): Let $U$ be an open subset of $X$ and let $\{\mathord{\uparrow } F_{i}:i\in I\}\subseteq \mathbf{Fin} X$ satisfy

\begin{equation*} C:=\bigcap \{\mathord {\uparrow } F_i : i\in I\}\subseteq U .\end{equation*}

As $I_0$ varies over the finite subsets of $I$ , the sets $\bigcap _{i\in I_0} \uparrow F_i$ form a filtered family of closed sets in the $\omega$ -topology with intersection $C$ . By the hypothesized well-filtering property (1), one of these sets must be contained in $U$ , and thus property R holds.

(2) $\Rightarrow$ (1): Let $U$ be an open subset of $X$ and let $\mathscr{A}$ be a filtered family of sets closed in the $\omega$ -topology such that $\bigcap _{A\in \mathscr{A}} A\subseteq U$ . Consider the family

\begin{equation*} \mathscr {F}:= \{\mathord {\uparrow } F: \vert F\vert \lt \infty, \,A\subseteq \mathord {\uparrow } F \textrm { for some }A\in \mathscr {A}\}. \end{equation*}

Each $A\in \mathscr{A}$ is the intersection of members of $\mathscr{F}$ since the sets $\mathord{\uparrow } F$ , $F$ finite, form a basis for the closed sets of the $\omega$ -topology, and thus $\bigcap \mathscr{F}=\bigcap \mathscr{A}\subseteq U$ . By property R there exists finitely many members $\mathord{\uparrow } F_1,\ldots, \mathord{\uparrow } F_n$ of $\mathscr{F}$ such that $\bigcap _{i=1}^n \mathord{\uparrow } F_i\subseteq U$ . By choice of the $F_i$ we may pick for each $i$ some $A_i\in \mathscr{A}$ such that $A_i\subseteq \mathord{\uparrow } F_i$ . By the filteredness of $\mathscr{A}$ , we may pick $A\in \mathscr{A}$ such that $A\subseteq \bigcap _{i=1}^n A_i$ . Then $A\subseteq U$ , and hence the well-filtering property is established.

(2) $\Leftrightarrow$ (3): The complements of the sets $\mathord{\uparrow } F$ , $F$ finite, form a basis for the $\omega$ -topology. By taking compliments, we can read property R to say any open cover of a closed set $A=X\setminus U$ by such basic open sets has a finite subcover. Equivalently by the Alexander Subbasis Theorem, the closed set $A$ is compact in the $\omega$ -topology.

(2) $\Leftrightarrow$ (4): The property of (4) is essentially property R restricted to sets of the form $\mathord{\uparrow } x$ instead of $\mathord{\uparrow } F$ , $F$ finite. These sets form a subbasis for the $\omega$ -closed sets, so the proof follows along the lines of (2) $\Leftrightarrow$ (3).

(1) $\Rightarrow$ (5): It follows directly from (1) that any basic open set $\Box U$ in the upper Vietoris topology on $\omega ^*(X)$ is Scott-open in the lattice $\mathscr{Q}$ .

(5) $\Rightarrow$ (2): It is straightforward to deduce property R from the hypothesis that each $\Box U$ is Scott-open in the lattice $\mathscr{Q}$ .

We can enhance the preceding slightly if we are working with dcpos equipped with the Scott topology.

Recall that for a topological space $(X, \tau )$ , the de Groot dual $\tau ^d$ of $\tau$ is defined by taking as a subbasis for the closed sets all compact saturated sets in $(X, \tau )$ . The patch topology $\tau ^{\sharp }$ on $X$ is the coarsest topology that is finer than the original topology $\tau$ and its de Groot dual $\tau ^d$ , namely, $\tau ^{\sharp }=\tau \bigvee \tau ^d$ .

See Example 56 of the next section for an example of the Scott space of a dcpo (indeed an algebraic domain) that does not satisfy property R (and hence the equivalent properties).

The following corollary follows directly from Proposition 20 and the Theorem35(4) by taking directed sets for $S$ .

A significant question in the study of $T_0$ -spaces equipped with the lower topology is the identification of useful conditions for the Lawson topology induced by the given topology, the topology generated by the given topology and the lower topology, to be compact. Using property R we can specify necessary and sufficient conditions.

We can modify the preceding ideas to derive a sufficient condition for a space to be an R-space.

We recall a result important for our purposes from Lawson et al. (Reference Lawson, Wu and Xi2020, Theorem 7.1).

From the preceding Lemma 25 and Jia et al. (Reference Jia2018, Theorem 3.4), we derive the following equivalences.

Now we turn to locally hypercompact spaces (see Definition 15), also called locally finitary compact spaces (Goubault-Larrecq Reference Goubault-Larrecq2013, Exercise 5.1.42) or $qc$ -spaces.

We refer the reader to Lawson (Reference Lawson1998) for results related to the previous lemma and following proposition.

4. Strong $\textit{d}$ -Spaces

As a strengthened version of $d$ -spaces, the notion of strong $d$ -spaces was introduced in Xu and Zhao (Reference Xu and Zhao2020). In this section, we will give some characterizations of strong $d$ -spaces. These characterizations indicate that the notion of a strong $d$ -space is, in some sense, a variant of join continuity. We also find conditions on a $d$ -space $X$ under which $X$ is sober. In particular, we show that for a dcpo $P$ , if $\Sigma P$ is a strong $d$ -space and $\Sigma (P\times P)=\Sigma P\times \Sigma P$ , then $\Sigma P$ is sober.

We list some elementary properties of strong $d$ -spaces.

Fig. 1 shows certain relations of some spaces lying between $d$ -spaces and $T_2$ -spaces (all implications in Fig. 1 are irreversible).

Figure 1.

Relations of some spaces lying between $d$ -spaces and $T_2$ spaces.

In Li et al. (Reference Li, Jin, Miao and Chen2023, Example 5.2), a poset $P$ was given to show that $P$ equipped with a certain topology $\tau$ is a strong $d$ -space but the product space $(P, \tau )\times (P, \tau )$ is not a strong $d$ -space (and hence the category $\mathbf{S}$ - $\mathbf{Top}_d$ is not a reflective subcategory of $\mathbf{Top}_0$ ). Using this space we will show that the product of two R-spaces is not an R-space in general.

Example 51. Let $P=\mathbb{N}\cup \{\omega \}\cup \{\beta _1, \beta _2, \ldots, \beta _n, \ldots \}\cup \{a_1, a_2, \ldots, a_n, \ldots \}\cup \{a\}$ . Define an order on $P$ as follows (see Fig. 2 ):

1. (i) $1\lt 2\lt 3\lt \ldots \lt n\lt n+1\lt \omega \textrm{ for all }n\in \mathbf{N}$ ;

2. (ii) $\omega \lt \beta _n \textrm{ for all }n\in \mathbf{N}$ ;

3. (iii) $n\lt a_m \textrm{ iff }n\leq m$ ;

4. (iv) $a\lt \beta _n \textrm{ and } a\lt a_n \textrm{ for all }n\in \mathbf{N}$ .

Figure 2.

The poset $P$ in Example 51.

Let $B=\{\beta _1, \beta _2, \ldots, \beta _n, \ldots \}$ and $A=\{a_1, a_2, \ldots, a_n, \ldots \}$ , and let $\tau _1=\{U\in \sigma (P) : A\setminus U \textrm{ is finite} \}$ and $\tau _2=\{U\in \sigma (P) : U\subseteq A\}$ . Define $\tau =\tau _1\bigcup \tau _2$ . Then

1. (a) $\textrm{max}(P)=A\cup B$ .

2. (b) $D\in \mathscr{D}(P)$ iff $D\subseteq \mathbb{N}$ is an infinite chain or $D$ has a largest element. So $P$ is a dcpo. If $D\subseteq \mathbb{N}$ is an infinite chain or $D$ has a largest element, then $D\in \mathscr{D}(P)$ . Conversely, suppose that $D$ is a directed subset of $P$ and $D$ has no largest element. Then $D$ is countably infinite and $|D\cap \textrm{max}(P)|\lt 2$ . If there exists $d^{\ast }\in D$ with $d^{\ast }\in \textrm{max}(P)$ , then for any $d\in D$ , by the directedness of $D$ , there is $d^{\prime }\in D$ such that $d\leq d^{\prime }$ and $d^{\ast }\leq d^{\prime }$ . Then $d\leq d^{\prime }=d^{\ast }$ . So $d^{\ast }$ is the largest element of $D$ , a contradiction. Hence, $D\cap \textrm{max}(P)=\emptyset$ , that is, $D\subseteq \mathord{\downarrow } \omega \cup \{a\}$ . Since for any $s\in \mathord{\downarrow } \omega$ , $s$ and $a$ have no upper bound in $D$ , we have $a\not \in D$ , and hence, $D\subseteq \mathord{\downarrow } \omega$ . As $D$ has no largest element, $D\subseteq \mathbb{N}$ is an infinite chain.

3. (c) $P$ is an algebraic domain. By (b), $x\ll x$ for all $x\in P\setminus \{\omega \}$ . Hence, $P$ is an algebraic domain.

4. (d) $\Sigma P$ is sober, not a strong $d$ -space and not coherent. By (c) and Proposition 22, $\Sigma P$ is sober. We have that $\bigcap _{n\in \mathbb{N}}\mathord{\uparrow } n\cap \mathord{\uparrow } a=\{\beta _1, \beta _2, \ldots, \beta _n, \ldots \}\in \sigma (P)$ , but $\mathord{\uparrow } m\cap \mathord{\uparrow } a=\{\beta _1, \beta _2, \ldots, \beta _n, \ldots \}\cup \{a_m, a_{m+1}, \ldots \}\not \subseteq \{\beta _1, \beta _2, \ldots, \beta _n, \ldots \}$ for any $m\in \mathbb{N}$ . Hence, $\Sigma P$ is not a strong $d$ -space. Clearly, $\mathord{\uparrow } 1, \mathord{\uparrow } a\in \mathord{\mathsf{K}} (\Sigma P)$ , but $\mathord{\uparrow } 1\cap \mathord{\uparrow } a=A\cup B$ is not Scott-compact (note that $\{\beta _n\}, \{a_n\}\in \sigma (P)$ for any $n\in \mathbb{N}$ ). So $\Sigma P$ is not coherent.

5. (e) $\tau$ is a topology on $P$ and $\upsilon (P)\subseteq \tau \subseteq \sigma (P)$ . Hence, $\tau$ is an order-compatible topology of $P$ and $(P, \tau ))$ is a $d$ -space. It was showed in (Li et al., 2023, Example 5.2).

6. (f) $(P, \tau )$ is sober and not coherent. Suppose that $C\in{\mathsf{Irr}}_c((P, \tau ))$ . Then by (e) and Lemma 19, $C=\mathord{\downarrow } \textrm{max}(P)$ . We claim that $\textrm{max}(C)$ is finite. Assume, on the contrary, that $\textrm{max}(C)$ is (countably) infinite. Since $C$ is Scott-closed and $\omega =\bigvee \mathbb{N}$ , $\textrm{max}(C)\cap (\mathord{\downarrow } \omega \cup \{a\})$ is a finite set, and consequently, $\textrm{max}(C)\cap \textrm{max}(P)$ is infinite.

   Case 1: $|\textrm{max}(C)\cap A|\geq 2$ . Select any $a_l, a_k\in \textrm{max}(C)\cap A$ with $a_l\neq a_k$ . Let $U_1=\{a_l\}$ and $U_2=\{a_k\}$ . Then $U_1, U_2\in \tau _2\subseteq \tau$ , $a_l\in C\cap U_1$ and $a_k\in C\cap U_2$ . But $A\cap U_1\cap U_2=\emptyset$ , which is a contradiction with $C\in{\mathsf{Irr}}_c((P, \tau ))$ .

   Case 2: $|\textrm{max}(C)\cap A|\lt 2$ . As $\textrm{max}(C)\cap \textrm{max}(P)=(\textrm{max}(C)\cap A)\cup (\textrm{max}(C)\cap B)$ is infinite, $\textrm{max}(C)\cap B$ must be infinite. Select any $\beta _n, \beta _m\in \textrm{max}(C)\cap B$ with $\beta _n\neq \beta _m$ . Let $V_1=\{\beta _n\}\cup (A\setminus \textrm{max}(C)\cap A)$ and $V_2=\{\beta _m\}\cup (A\setminus \textrm{max}(C)\cap A)$ . Then $V_1, V_2\in \tau _1\subseteq \tau$ , $\beta _n\in C\cap V_1$ and $\beta _m\in C\cap V_2$ . But $A\cap V_1\cap V_2=\emptyset$ , which is in contradiction with $C\in{\mathsf{Irr}}_c((P, \tau ))$ . Hence, $\textrm{max}(C)=\{x_1, x_2, x_3, \ldots, x_n\}$ is finite, then $C=\mathord{\downarrow } \textrm{max}(C)=\bigcup \limits _{i=1}^n\mathord{\downarrow } x_i$ and hence $C=\mathord{\downarrow } x_m={\textrm{cl}}_{\tau }\{x_m\}$ for some $1\leq m\leq n$ by $C\in{\mathsf{Irr}}_c((P, \tau ))$ . Thus, $(P, \tau ))$ is sober. Clearly, $\mathord{\uparrow } 1, \mathord{\uparrow } a\in \mathord{\mathsf{K}} (\Sigma P)$ , but $\mathord{\uparrow } 1\cap \mathord{\uparrow } a=A\cup B$ is not compact (note that $\{\{\beta _n\}\cup A : n\in \mathbb{N}\}$ is a $\tau$ -open cover of $A\cup B$ containing no finite subcover). Therefore, $(P, \tau ))$ is not coherent.

7. (g) $(P, \tau ))$ is an R-space and hence a strong $d$ -space. Suppose that $S$ is a subset of $P$ and $U\in \tau$ with $\bigcap _{s\in S}\mathord{\uparrow } s\subseteq U$ . We will show that there is a finite subset $S_0$ of $S$ such that $\bigcap _{s\in S_0}\mathord{\uparrow } s\subseteq U$ . If $S$ itself is finite, then we let $S_0=S$ . Now we assume that $S$ is (countably) infinite.

   Case 1: $|S\cap \textrm{max}(P)|\geq 2$ . Select any $s_1, s_2\in S\cap \textrm{max}(P)$ with $s_l\neq s_2$ . Then $\mathord{\uparrow } s_1\cap \mathord{\uparrow } s_2=\emptyset \subseteq U$ .

   Case 2: $|S\cap A|=1$ and $S\cap B=\emptyset$ . In this case, $S\cap \textrm{max}(P)=\{a_m\}$ for some $m\in \mathbb{N}$ and $S\cap \mathbb{N}$ is infinite. Select any $k\in S\cap \mathbb{N}$ with $m\lt k$ . Then $\mathord{\uparrow } k\cap \mathord{\uparrow } a_m=\emptyset \subseteq U$ .

   Case 3: $|S\cap B|=1$ and $S\cap A=\emptyset$ . Then $S\cap \textrm{max}(P)=\{\beta _l\}$ for some $l\in \mathbb{N}$ and $S\subseteq \mathord{\downarrow } \omega \cup \{\beta _l\}\cup \{a\}$ . Select any $s\in S\cap \mathord{\downarrow } \omega$ , Then $\mathord{\uparrow } s\cap \mathord{\uparrow } \beta _l=\bigcap _{s\in S\cap \mathord{\downarrow } \omega }\mathord{\uparrow } s\cap \mathord{\uparrow } \beta _l\cap \mathord{\uparrow } a=\bigcap _{s\in S}\mathord{\uparrow } s=\{\beta _l\}\subseteq U$ .

   Case 4: $S\cap \textrm{max}(P)=\emptyset$ . We have that $S\subseteq \mathord{\downarrow } \omega \cup \{a\}$ . If $a\in S$ , then $\bigcap _{s\in S}\mathord{\uparrow } s=B\subseteq U$ . So $A\setminus U$ is finite, and consequently, there is $k\in \mathbb{N}$ such that $\{a_k, a_{k+1}, \ldots \}\subseteq U$ . Select any $m\in S\cap \mathbb{N}$ with $m\gt k$ (note that $S\cap \mathbb{N}$ is infinite). Then $\mathord{\uparrow } m\cap \mathord{\uparrow } a=B\cup \{a_m, a_{m+1}, \ldots \}\subseteq B\cup \{a_k, a_{k+1}, \ldots \}\subseteq U$ . If $a\not \in S$ , then $\bigcap _{s\in S}\mathord{\uparrow } s=\mathord{\uparrow } \omega =B\cup \{\omega \}\subseteq U$ . By $\omega =\bigvee (S\cap \mathbb{N})$ and $U\in \tau \subseteq \sigma (P)$ , there is $n\in S\cap \mathbb{N}$ with $\mathord{\uparrow } n\subseteq U$ . Thus, by Theorem 35 $(P, \tau ))$ is an R-space.

8. (h) The product space $(P, \tau ))\times (P, \tau ))$ is not an R-space. It was proved in Li et al. (Reference Li, Jin, Miao and Chen2023, Example 5.2) that $(P, \tau ))\times (P, \tau ))$ is not a strong $d$ -space. By Lemma 50, $(P, \tau ))\times (P, \tau ))$ is not an R-space.

By Example 51 and Li et al. (Reference Li, Jin, Miao and Chen2023, Example 5.2), we pose the following question.

By Example 51 and MacLane (Reference MacLane1997, pp. 92, Exercise 7) (or Nel and Wilson Reference Nel and Wilson1972, Remark 1.1), we get the following result.

For a dcpo $P$ , $(P, \upsilon (P))$ and $(P, \sigma (P))$ need not be strong $d$ -spaces, although they are always $d$ -spaces. Consider the Johnstone’s dcpo $\mathbb{J}=\mathbb{N}\times (\mathbb{N}\cup \{\infty \})$ with ordering defined by $(m,p)\leq (n,q)$ if $m=n$ and $p\leq q$ or if $p\leq n$ and $q=\infty$ . Then $(\mathbb{J}, \upsilon (\mathbb{J}))$ and the Johnstone space $\Sigma \,\!\mathbb{J}$ are $d$ -spaces. Clearly, $\bigcap _{n\in \mathbb{N}}\mathord{\uparrow } (1, n)\cap \mathord{\uparrow } (2,1)=\emptyset$ , but $\mathord{\uparrow } (1, n)\cap \mathord{\uparrow } (2,1)=\{(m, \omega ): n\leq m\}\neq \emptyset$ for all $n$ . Hence, $(\mathbb{J}, \upsilon (\mathbb{J}))$ and $\Sigma \,\!\mathbb{J}$ are not strong $d$ -spaces. Example 58 below shows that there is even an algebraic domain $P$ (hence $\Sigma P$ is sober) such that $\Sigma P$ is not a strong $d$ -space.

The following example shows that a locally compact and second-countable strong $d$ -space may not be a well-filtered space in general (and hence not sober). So a locally compact strong $d$ -space may not be sober (in contrast to Theorem29) and a second-countable strong $d$ -space need not be sober (in contrast to Xu et al. Reference Xu, Shen, Xi and Zhao2020a, Theorem 4.2).

The following result follows directly from Proposition 11(4).

In the following example, we give a Noetherian dcpo $P$ such that $\Sigma P$ is not an R-space, though it is both a strong $d$ -space and a sober space. It also shows that there exists a Noetherian dcpo $P$ such that $(P, \tau )$ is not an R-space for any order-compatible topology on $P$ (in contrast to Proposition 55).