Some Recent Advances in Non-Hausdorff Topology

Abstract

In the past few years, the research in non-Hausdorff topology, especially on sober spaces and well-filtered spaces, has got some breakthrough progress. In this paper, we shall present a brief summarising survey on some of such development. Some related problems are listed.

1. Introduction

In most topology books, the Hausdorff separation property is assumed from the very start and very little information is contained on non-Hausdorff spaces. In classical mathematics, most topological spaces are indeed Hausdorff. But non-Hausdorff spaces are important already in algebraic geometry, and crucial in fields such as domain theory. Indeed, in connection with order, non-Hausdorff spaces (especially $T_0$ spaces) play a more significant role than Hausdorff spaces.

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 very useful and important properties for non-Hausdorff topology¹ theory: $d$-space and well-filtered space (see [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52]). In the past few years, some remarkable progresses have been achieved in understanding such structures. In this chapter, we shall make a brief survey on some of this progress and list a few related problems.

2. Preliminary

We now recall some basic concepts and notations that will be used in the paper. For further details, we refer the reader to [4, 7, 53].

A partial order $\le$ on a set $X$ is a transitive (i.e., $x\le y$ and $y\le z$ imply $x\le z$), reflexive (it means $x\le x$ for any $x\in P$), and antisymmetric (i.e., $x\le y$ and $y\le x$ imply $x=y$) relation. A partially ordered set, or poset for short, is a nonempty set $P$ equipped with a partial order $\le$.

Let $P$ be a poset. For any $A\subseteq P$, let $\downarrow A=\left\{x\in P:x\le a\mathit{\text{for}}\mathit{\text{some}}a\in A\right\}$ and $\uparrow A=\left\{x\in P:x\ge a\mathit{\text{for}}\mathit{\text{some}}a\in A\right\}$. For each $x\in P$, we write $\downarrow x$ for $\downarrow \left\{x\right\}$ and $\uparrow x$ for $\uparrow \left\{x\right\}$. A subset $A$ of $P$ is called a lower set (resp., an upper set) if $A=\downarrow A$ (resp., $A=\uparrow A$). Define $P^{\left(<\omega \right)}=\left\{F\subseteq P:F\text{is}\mathrm{a}\text{nonempty}\text{finite}\mathrm{set}\right\}$ and $\mathbf{Fin}P=\left\{\uparrow F:F\in P^{\left(<\omega \right)}\right\}$. For a nonempty subset $A$ of $P$, define $max\left(A\right)=\left\{a\in A:a\text{is}\mathrm{a}\text{maximal}\text{element}\text{of}A\right\}$ and $min\left(A\right)=\left\{a\in A:a\text{is}\mathrm{a}\text{minimal}\text{element}\text{of}A\right\}$. The symbol $\mathbb{N}$ will denote the poset of all natural numbers in the usual order.

A nonempty subset $D$ of a poset $P$ is called directed if every two elements in $D$ have an upper bound in $D$. Let $\mathcal{D}\left(P\right)$ denote the set of all directed sets of $P$. A poset $P$ is called a directed complete poset, or dcpo for short, if the supremum $\vee D$ of $D$ exists in $P$ for every $D\in \mathcal{D}\left(P\right)$. A subset $I\subseteq P$ is called an ideal if $I$ is a directed lower subset of $P$. Let $\mathrm{Id}\left(P\right)$ denote the poset of all ideals of $P$ with the set inclusion order. Dually, we define the filters and denote the poset of all filters of $P$ by $\text{Filt}\left(P\right)$.

For a poset $Q$, the upper topology on $Q$, generated by $\left\{Q\backslash \downarrow x:x\in Q\right\}$ as a subbase, is denoted by $\upsilon \left(Q\right)$. Dually, we define the lower topology on $Q$ and is denoted by $\omega \left(Q\right)$. The upper sets of $Q$ form the (upper) Alexandroff topology $\alpha \left(Q\right)$.

Definition 2.1. For a poset $P$ and $U\subseteq P$, $U$ is said to be Scott open if

1. $U=\uparrow U$, and

2. for any directed subset $D$ for which $\vee D$ exists, $\vee D\in U$ implies $D\cap U\ne \varnothing$.

All Scott open subsets of $P$ form a topology, called the Scott topology on $P$ and denoted by $\sigma \left(P\right)$. The space $\mathrm{\Sigma }P=\left(P,\sigma \left(P\right)\right)$ is called the Scott space of $P$. The topology generated by $\omega \left(P\right)\cup \sigma \left(P\right)$ is called the Lawson topology on $P$ and is denoted by $\lambda \left(P\right)$. The space $\mathrm{\Lambda }P=\left(P,\lambda \left(P\right)\right)$ is called the Lawson space of $P$.

The following result is well-known (cf. [4], Proposition II-2.1).

Lemma 2.2. Let $P,Q$ be posets and $f:P\to Q$. Then the following two conditions are equivalent:

1. $f$ is Scott continuous, that is, $f:\mathrm{\Sigma }P\to \mathrm{\Sigma }Q$ is continuous.

2. For any $D\in \mathcal{D}\left(P\right)$ for which $\vee D$ exists, $f\left(\vee D\right)=\vee f\left(D\right)$.

For a $T_0$ space $X$, the specialization order ${\le }_X$ on $X$ is defined by $x{\le }_{X}y$ iff $x\in \overline{\left\{y\right\}}$). In the following, when a $T_0$ space is considered as a poset, the order shall mean the specialization order provided a different one is specified. Let $\mathcal{O}\left(X\right)$ (resp., $\mathcal{C}\left(X\right)$) be the set of all open subsets (resp., closed subsets) of space $X$. For $A\subseteq X$, the closure of $A$ in $X$ is denoted by $\mathrm{cl}A$, or simply by $\overline{A}$. Define ${\mathcal{S}}_c\left(X\right)=\left\{\overline{\left\{x\right\}}:x\in X\right\}$ and ${\mathcal{D}}_c\left(X\right)=\left\{\overline{D}:D\in \mathcal{D}\left(X\right)\right\}$. For two spaces $X$ and $Y$, we use the symbol $X\cong Y$ to denote that $X$ and $Y$ are homeomorphic.

A $T_0$ space $X$ is called a $d$-space (or monotone convergence space) if $X$ (with the specialization order) is a dcpo and $\mathcal{O}\left(X\right)\subseteq \sigma \left(X\right)$ (cf. [4, 32]). For any dcpo $P$, $\mathrm{\Sigma }P$ is clearly a $d$-space. The category of all $d$-spaces and continuous mappings is denoted by ${\mathbf{Top}}_d$.

For a $d$-space $X$ and a nonempty closed subset of $X$, if $x\in A$, then by Zorn’s Lemma there is a maximal chain $C_x$ in $A$ with $x\in C_x$. Since $X$ is a $d$-space, $c_x=\vee C_x$ exists and $c_x\in A$. By the maximality of $C_x$, we have $c_x\in max\left(A\right)$ and $x\le c_x$. Therefore, $A\subseteq \downarrow max\left(A\right)\subseteq \downarrow A=A$, and hence $A=\downarrow max\left(A\right)$. So we have the following.

Lemma 2.3. Let $X$ be a $d$-space. If $A$ is a nonempty closed subset of $X$, then $A=\downarrow max\left(A\right)$ and hence $max\left(A\right)\ne \varnothing$.

Lemma 2.4. ([39]) Let $P$ be a poset and $Y$ a $d$-space. Then for any $f:P\to Y$, the following two conditions are equivalent:

1. $f:\mathrm{\Sigma }P\to Y$ is continuous.

2. $f:\mathrm{\Sigma }P\to \mathrm{\Sigma }Y$ is continuous.

A nonempty subset $A$ of a $T_0$ space $X$ is called irreducible if for any $\left\{F_1,F_2\right\}\subseteq \mathcal{C}\left(X\right)$, $A\subseteq F_1\cup F_2$ implies $A\subseteq F_1$ or $A\subseteq F_2$. We denote by $\mathrm{Irr}\left(X\right)$ (resp., ${\mathrm{Irr}}_c\left(X\right)$) the set of all irreducible (resp., irreducible closed) subsets of $X$. Clearly, every subset of $X$ that is directed under ${\le }_X$ is irreducible. The space $X$ is called sober, if for any $F\in {\mathrm{Irr}}_c\left(Y\right)$, there is a (unique) point $a\in X$ such that $F=\overline{\left\{a\right\}}$. The category of all sober spaces and continuous mappings is denoted by $\mathbf{Sob}$.

For a family $X_i\left(i\in I\right)$ of $T_0$ spaces and the product space $X={\prod }_{i\in I}{X}_i$, let $p_i:X\to X_i$ ($i\in I$) be the $i$ th projection.

Lemma 2.5. ([42]) Let $X={\prod }_{i\in I}{X}_i$ be the product space of $T_0$ spaces $X_i\left(i\in I\right)$. If $A\in {\mathrm{Irr}}_c\left(X\right)$, then $A={\prod }_{i\in I}{p}_i\left(A\right)$ and $p_i\left(A\right)\in {\mathrm{Irr}}_c\left(X_i\right)$ for each $i\in I$.

A subset $A$ of a $T_0$ space $X$ is called saturated if $A$ equals the intersection of all open sets containing it (equivalently, $A$ is an upper set with respect to the specialization order). We use $\mathtt{K}\left(X\right)$ to denote the set of all nonempty compact saturated subsets of a $T_0$ space $X$ and endow $\mathtt{K}\left(X\right)$ with the Smyth order, that is, for $K_1,K_2\in \mathtt{K}\left(X\right)$, $K_1⊑K_2$ iff $K_2\subseteq K_1$. Let $\text{OFilt}\left(\mathcal{O}\left(\mathrm{X}\right)\right)=\sigma \left(\mathcal{O}\left(X\right)\right)\cap \text{Filt}\left(\mathcal{O}\left(X\right)\right)$. The members of $\text{OFilt}\left(\mathcal{O}\left(\mathrm{X}\right)\right)$ are called open filters of $X$. For each $K\in \mathtt{K}\left(X\right)$, let $\mathrm{\Phi }\left(K\right)=\left\{U\in \mathcal{O}\left(X\right):K\subseteq U\right\}$. Then $\mathrm{\Phi }\left(K\right)\in \text{OFilt}\left(\mathcal{O}\left(\mathrm{X}\right)\right)$ and $K=\cap \mathrm{\Phi }\left(K\right)$. Obviously, $\mathrm{\Phi }:\mathtt{K}\left(X\right)\to \text{OFilt}\left(\mathcal{O}\left(\mathrm{X}\right)\right),K\mapsto \mathrm{\Phi }\left(K\right)$, is an order embedding.

For a $T_0$ space $X$, $\mathcal{G}\subseteq 2^X$ and $W\subseteq X$, let ${\diamond }_{\mathcal{G}}W=\left\{G\in \mathcal{G}:G\cap W\ne \varnothing \right\}$ and ${\square }_{\mathcal{G}}W=\left\{G\in \mathcal{G}:G\subseteq A\right\}$. The symbols ${\diamond }_{\mathcal{G}}W$ and ${\square }_{\mathcal{G}}W$ will be simply written as $\diamond A$ and $\square A$ respectively if no ambiguity occur. The lower Vietoris topology on $\mathcal{G}$ is the topology that has $\left\{\diamond U:U\in \mathcal{O}\left(X\right)\right\}$ as a subbase, and the resulting space is denoted by $P_H\left(\mathcal{G}\right)$. The space $P_H\left(\mathcal{C}\left(X\right)\backslash \left\{\varnothing \right\}\right)$, $P_H\left(X\right)$ for short, is called the Hoare power space or lower space of $X$ (cf. [27]). Clearly, $P_H\left(X\right)=\left(\mathcal{C}\left(X\right)\backslash \left\{\varnothing \right\},\upsilon \left(\mathcal{C}\left(X\right)\backslash \left\{\varnothing \right\}\right)\right)$, and hence $P_H\left(X\right)$ is always sober (see [51], Corollary 4.10) or [42], Proposition 2.9). The upper Vietoris topology on $\mathtt{K}\left(X\right)$ is the topology generated by $\left\{\square U:U\in \mathcal{O}\left(X\right)\right\}$ as a base, and the resulting space is called the Smyth power space or upper space of $X$ and is denoted by $P_S\left(X\right)$ (cf. [8, 9, 27]).

Remark 2.6. Let $X$ be a $T_0$ space.

1. If ${\mathcal{S}}_c\left(X\right)\subseteq \mathcal{G}$, then the specialization order on $P_H\left(\mathcal{G}\right)$ is the set inclusion order, and the canonical mapping ${\eta }_X:X\to P_H\left(\mathcal{G}\right)$, given by ${\eta }_X\left(x\right)=\overline{\left\{x\right\}}$, is a topological embedding (cf. [4, 7, 27]).

2. The space $X^s=P_H\left({\mathrm{Irr}}_c\left(X\right)\right)$ with the canonical mapping ${\eta }_X:X\to X^s$ is the (standard) sobrification of $X$ (cf. [4, 7]).

3. the specialization order on $P_S\left(X\right)$ is the Smyth order, that is, ${\le }_{P_S\left(X\right)}=⊑$;

4. the canonical mapping ${\xi }_X:X\to P_S\left(X\right)$, $x\mapsto \uparrow x$, is an order and topological embedding (cf. [8, 9, 27])

For a nonempty subset $C$ of a $T_0$ space $X$, it is easy to see that $C$ is compact iff $\uparrow C\in \mathtt{K}\left(X\right)$. Furthermore, we have the following useful result (see, e.g., [10]).

Lemma 2.7. Let $X$ be a $T_0$ space and $C\in \mathtt{K}\left(X\right)$. Then $C=\uparrow min\left(C\right)$ and $min\left(C\right)$ is compact.

The Smyth power space construction defines a covariant functor. More precisely, we have the following.

Lemma 2.8. ([37]) $P_S:{\mathbf{Top}}_0\to {\mathbf{Top}}_0$ is a covariant functor, where for any $f:X\to Y$ in ${\mathbf{Top}}_0$, $P_S\left(f\right):P_S\left(X\right)\to P_S\left(Y\right)$ is defined by $P_S\left(f\right)\left(K\right)=\uparrow f\left(K\right)$ for all $K\in \mathtt{K}\left(X\right)$.

A $T_0$ space $X$ is called well-filtered if for any open set $U$ and any $\mathcal{K}\in \mathcal{D}\left(\mathtt{K}\left(X\right)\right)$, $\cap \mathcal{K}\subseteq U$ implies $K\subseteq U$ for some $K\in \mathcal{K}$. The category of all well-filtered spaces and continuous mappings is denoted by ${\mathbf{Top}}_w$.

As in [54], a topological space $X$ is locally hypercompact (resp., a $C$-space) if for each $x\in X$ and each open neighborhood $U$ of $x$, there is $\uparrow F\in \mathbf{Fin}X$ such that $x\in int\uparrow F\subseteq \uparrow F\subseteq U$ (resp., there is $u\in X$ such that $x\in int\uparrow u\subseteq \uparrow u\subseteq U$.). A set $K\subseteq X$ is called supercompact if for any family $\left\{U_i:i\in I\right\}\subseteq \mathcal{O}\left(X\right)$, $K\subseteq {\cup }_{i\in I}{U}_i$ implies $K\subseteq U$ for some $i\in I$. It is easy to verify that the supercompact saturated sets of $X$ are exactly the sets $\uparrow x$ with $x\in X$ (see [9], Fact 2.2). It is well-known that $X$ is a $C$-space iff $\mathcal{O}\left(X\right)$ is a completely distributive lattice (cf. [55]). A space $X$ is called core compact if $\left(\mathcal{O}\left(X\right),\subseteq \right)$ is a continuous lattice (cf. [4]).

For a full subcategory $\mathbf{K}$ of ${\mathbf{Top}}_0$ and an object $X$ of $\mathbf{K}$, we will call $X$ a $\mathbf{K}$-space. The category $\mathbf{K}$ is said to be a Keimel-Lawson category if the following four properties are satisfied:

(${\mathrm{K}}_1$) Homeomorphic copies of $\mathbf{K}$-spaces are $\mathbf{K}$-spaces.

(${\mathrm{K}}_2$) $\mathbf{Sob}\subseteq \mathbf{K}$, that is, all sober spaces are $\mathbf{K}$-spaces.

(${\mathrm{K}}_3$) In a sober space $X$, the intersection of any family of $\mathbf{K}$-subspaces of $X$ is a $\mathbf{K}$-space.

(${\mathrm{K}}_4$) Continuous maps $f:S\to T$ between sober spaces $S$ and $T$ are $\mathbf{K}$-continuous, that is, for every $\mathbf{K}$-subspace $K$ of $T$, the inverse image $f^{-1}\left(K\right)$ is a $\mathbf{K}$-subspace of $S$.

In what follows, $\mathbf{K}$ always refers to a full subcategory of ${\mathbf{Top}}_0$ containing $\mathbf{Sob}$. The category $\mathbf{K}$ is said to be closed with respect to homeomorphisms if $\mathbf{K}$ has (${\mathrm{K}}_1$).

3. Well-filtered spaces and sober spaces

It is well-known that every sober space is well-filtered (see [14]) and every well-filtered space is a $d$-space (cf. [33, 42]). The Scott space of every continuous dcpo is sober (see [4]). Furthermore, the Scott space of every quasicontinuous domain is sober (see [6]). Johnstone [17] constructed the first dcpo whose Scott space is non-sober. Soon after, Isbell [15] gave a complete lattice whose Scott space is non-sober. The general problem in this line is whether each object in a classic class of dcpos or complete lattices has a sober Scott space.

Example 3.1. (Johnstone’s dcpo) Let $\mathbb{J}=\mathbb{N}\times \left(\mathbb{N}\cup \left\{\infty \right\}\right)$ with ordering defined by $\left(j,k\right)\le \left(m,n\right)$ iff $j=m$ and $k\le n$, or $n=\infty$ and $k\le m$ (see Figure 1).

Figure 1.

Johnstone’s dcpo $\mathbb{J}$.

$\mathbb{J}$ is a well-known dcpo constructed by Johnstone in [17]. Now, we show that $\mathbb{J}$ is not well-filtered. Suppose, on the contrary, that there exists a well-filtered topology $\tau$ on $\mathbb{J}$ which is compatible with the original order of $\mathbb{J}$. Clearly, ${\cap }_{n\in \mathbb{N}}\left(\uparrow \left(1,n\right)\cap \uparrow \left(2,1\right)\right)=\varnothing$, and $\uparrow \left(1,n\right)\cap \uparrow \left(2,1\right)=\left\{\left(m,\infty \right):n\le m\right\}$ is a compact saturated subset in $\mathrm{\Sigma }\mathbb{J}$, and hence a compact saturated subset in $\left(\mathbb{J},\tau \right)$ since $\tau \subseteq \sigma \left(\mathbb{J}\right)$. By the well-filteredness of $\left(\mathbb{J},\tau \right)$, $\uparrow \left(1,n_0\right)\cap \uparrow \left(2,1\right)=\varnothing$ for some $n_0\in \mathbb{N}$, a contradiction. Thus, $\mathbb{J}$ is not well-filtered. In particular, $\mathrm{\Sigma }\mathbb{J}$ is not well-filtered (see [7], Exercise 8.3.9).

In 1992, Heckmann [8] asked whether every well-filtered dcpo is sober in its Scott topology. Kou [20] constructed the first dcpo whose Scott space is well-filtered but non-sober (see [16, 33, 52] for other different counterexamples) and hence gave a negative answer to Heckmann’s question.

In [33], Xi and Lawson found out a condition under which a $d$-space is well-filtered.

Theorem 3.2. ([33]) Let $X$ be a $d$-space. If $\downarrow \left(A\cap K\right)$ is closed for any $A\in \mathcal{C}\left(X\right)$ and $K\in \mathtt{K}\left(X\right)$, then $X$ is well-filtered.

Corollary 3.3. ([33]) For a dcpo $P$, if $\mathrm{\Lambda }P$ is compact (in particular, $P$ is a complete lattice), then $\mathrm{\Sigma }P$ is well-filtered.

It follows from Theorem 3.2 or Corollary 3.3 that Isbell’s non-sober complete lattice is well-filtered. Note that Johnstone’s dcpo is countable and the Isbell’s non-sober complete lattice $L$ is neither distributive nor countable. Thus in 1994, Abramsky and Jung asked whether there is a distributive complete lattice whose Scott space is non-sober (see [1], Exercises 7.3.19–6) or [18]). In [44], using Isbell’s lattice, Xu, Xi, and Zhao gave a positive answer to this problem.

Theorem 3.4. ([44]) Let $L$ be the Isbell’s non-sober lattice. Then $\mathtt{K}\left(\mathrm{\Sigma }L\right)$ is a spatial frame and the Scott space of $\mathtt{K}\left(\mathrm{\Sigma }L\right)$ is non-sober.

For the lattice of closed subsets of $\mathrm{\Sigma }L$, we have the following.

Proposition 3.5. ([25]) Let $L$ be the Isbell’s lattice. Then $\mathrm{\Sigma }L$ is a retract of $\mathrm{\Sigma }\mathcal{C}\left(\mathrm{\Sigma }L\right)$ and hence $\mathrm{\Sigma }\mathcal{C}\left(\mathrm{\Sigma }L\right)$ is non-sober.

In [56], Erné showed that for the complete Boolean algebra $B$ of all regular open subsets of the reals, the Scott space $\mathrm{\Sigma }B$ is not a topological join-semilattice (and hence the Scott topology $\sigma \left(B\times B\right)$ is strictly finer than the product topology $\sigma \left(B\right)\times \sigma \left(B\right)$). It is natural to wonder whether $\mathrm{\Sigma }B$ is sober. Thus Erné asked the following.

Question 3.6. Let $B$ be the complete Boolean algebra of all regular open subsets of the reals. Is the Scott space $\mathrm{\Sigma }B$ sober?

If the answer is negative, we would have a natural complete Boolean algebra whose Scott space is non-sober.

More generally, we pose the following question.

Question 3.7. ([47]) Is there a complete Boolean algebra $B$ such that $\mathrm{\Sigma }B$ is not sober?

In 2019, in a talk [18] given at the National Institute of Education, Singapore, Achim Jung also asked whether there is a countable complete lattice whose Scott space is non-sober.

In [26], Miao, Xi, Li, and Zhao gave a negative answer to this question.

Theorem 3.8. ([26]) There is a countably infinite complete lattice whose Scott space is non-sober.

One of the most important results on sober spaces is the Hofmann-Mislove Theorem (see [14], Theorem 2.16) or [4], Theorem II-1.20 and Theorem II-1.21).

Theorem 3.9. (Hofmann-Mislove Theorem) For a $T_0$ space $X$, the following conditions are equivalent:

1. $X$ is a sober space.

2. For any $\mathcal{F}\in \text{OFilt}\left(\mathcal{O}\left(X\right)\right)$, there is a $K\in \mathtt{K}\left(X\right)\cup \left\{\varnothing \right\}$ such that $\mathcal{F}=\mathrm{\Phi }\left(K\right)$.

3. For any $\mathcal{F}\in \text{OFilt}\left(\mathcal{O}\left(X\right)\right)$, $\mathcal{F}=\mathrm{\Phi }\left(\cap \mathcal{F}\right)$.

By Hofmann-Mislove Theorem, $\mathrm{\Phi }:\mathtt{K}\left(X\right)\cup \left\{\varnothing \right\}\to \text{OFilt}\left(\mathcal{O}\left(\mathrm{X}\right)\right)$ is an order isomorphism if and only if $X$ is sober.

In [16], Jia asked whether every core compact well-filtered space is sober. This problem was independently answered in [21] and in [41, 42] by different methods.

Theorem 3.10. ([21, 41, 42]) Every core compact well-filtered space is sober.

In addition, the following result was obtained by Xu et al. in [41].

Theorem 3.11. ([41]) Every first countable well-filtered $T_0$ space is sober.

By ([20], Theorem 2.3), ([4], Theorem V-5.6), and Theorem 3.10, one get the following important result.

Theorem 3.12. 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.

4. Rudin sets and well-filtered determined sets

Rudin’s Lemma is a very useful tool in non-Hausdorff topology and plays a crucial role in domain theory (see [1, 4, 7, 8]). The following topological variant of Rudin’s Lemma was given in [9].

Lemma 4.1. (Topological Rudin Lemma) Let $X$ be a topological space and $\mathcal{A}$ an irreducible subset of the Smyth power space $P_S\left(X\right)$. Then every closed set $C\subseteq X$ that meets all members of $\mathcal{A}$ contains a minimal irreducible closed subset $A$ that still meets all members of $\mathcal{A}$.

For a $T_0$ space $X$ and $\mathcal{K}\subseteq \mathtt{K}\left(X\right)$, let $M\left(\mathcal{K}\right)=\left\{A\in \mathcal{C}\left(X\right):K\cap A\ne \varnothing \text{for}\mathrm{all}K\in \mathcal{K}\right\}$ (that is, $\mathcal{A}\subseteq \diamond A$) and $m\left(\mathcal{K}\right)=\left\{A\in \mathcal{C}\left(X\right):A\text{is}\mathrm{a}\text{minimal}\text{menber}\text{of}M\left(\mathcal{K}\right)\right\}$.

Definition 4.2. ([42]) A $T_0$ space $X$ is called a directed closure space, $\mathrm{DC}$ space for short, if ${\mathrm{Irr}}_c\left(X\right)={\mathcal{D}}_c\left(X\right)$, that is, for each $A\in {\mathrm{Irr}}_c\left(X\right)$, there exists a directed subset of $X$ such that $A=\overline{D}$.

Remark 4.3. In [42], it was shown that closed subspaces, retracts and products of DC spaces are again DC spaces.

Based on the topological Rudin Lemma, the concept of Rudin spaces can be defined (see [29, 42]).

Definition 4.4. Let $X$ be a $T_0$ space and $A$ a nonempty subset of $X$.

1. $A$ is said to have the Rudin property, if there exists a filtered family $\mathcal{K}\subseteq \mathtt{K}\left(X\right)$ such that $\overline{A}\in m\left(\mathcal{K}\right)$ (that is, $\overline{A}$ is a minimal closed set that intersects all members of $\mathcal{K}$). Let $\mathtt{RD}\left(X\right)=\left\{A\in \mathcal{C}\left(X\right):A\mathrm{has}\text{Rudin}\text{property}\right\}$. The sets in $\mathtt{RD}\left(X\right)$ will also be called Rudin sets.

2. $X$ is called a Rudin space, RD space for short, if ${\mathrm{Irr}}_c\left(X\right)=\mathtt{RD}\left(X\right)$, that is, every irreducible closed set of $X$ is a Rudin set.

It was proved in [29] that the closed subspaces, retracts and product of Rudin spaces are again Rudin spaces.

Definition 4.5. ([36]) Let $\mathbf{K}$ be a full category of ${\mathbf{Top}}_0$ containing $\mathbf{Sob}$ and $X$ a $T_0$ space.

1. A subset $A$ of $X$ is said to be $\mathbf{K}$-determined, provided for any continuous mapping $f:X\to Y$ to a $\mathbf{K}$-space $Y$, there exists a unique $y_A\in Y$ such that $\overline{f\left(A\right)}=\overline{\left\{y_A\right\}}$. Denote by $\mathbf{K}\left(X\right)$ the set of all closed $\mathbf{K}$-determined sets of $X$.

2. The space $X$ is said to be a $\mathbf{K}$-determined space, if ${\mathrm{Irr}}_c\left(X\right)=\mathbf{K}\left(X\right)$ or, equivalently, all irreducible closed sets of $X$ are $\mathbf{K}$-sets.

For simplicity, let $d\left(X\right)={\mathbf{Top}}_d\left(X\right)$ and $\mathtt{WD}\left(X\right)={\mathtt{Top}}_w\left(X\right)$. The sets in $\mathtt{WD}\left(X\right)$ (resp., $d\left(X\right)$) are called WD sets (resp., $d$-determined sets). The space $X$ is called a well-filtered determined space, shortly a WD space, if all irreducible closed subsets of $X$ are WD sets, that is, ${\mathrm{Irr}}_c\left(X\right)=\mathtt{WD}\left(X\right)$ (see [36, 42]). The space $X$ is called a $d$-determined space if ${\mathrm{Irr}}_c\left(X\right)=d\left(X\right)$.

Proposition 4.6. ([36, 42]) Let $\mathbf{K}$ be a full category of ${\mathbf{Top}}_0$ containing $\mathbf{Sob}$ and $X$ a $T_0$ space. Then

1. ${\mathcal{S}}_c\left(X\right)\subseteq \mathbf{K}\left(X\right)\subseteq \mathbf{Sob}\left(X\right)={\mathrm{Irr}}_c\left(X\right)$.

2. ${\mathcal{S}}_c\left(X\right)\subseteq {\mathcal{D}}_c\left(X\right)\subseteq \mathtt{RD}\left(X\right)\subseteq \mathtt{WD}\left(X\right)\subseteq {\mathrm{Irr}}_c\left(X\right)$ and ${\mathcal{D}}_c\left(X\right)\subseteq d\left(X\right)\subseteq \mathtt{WD}\left(X\right)$.

3. Sober $\Rightarrow$ DC $\Rightarrow$ RD $\Rightarrow$ WD.

4. Every sober space is $\mathbf{K}$-determined.

From Proposition 4.6, we know that Rudin spaces lie between WD spaces and DC spaces. Also, DC spaces lie between Rudin spaces and sober spaces.

In ([22], Example 4.15), a $T_0$ space $X$ was constructed in which some well-filtered determined sets are not Rudin sets and hence gave a negative answer to a question posed by Xu and Zhao in [46]: Does $\mathtt{RD}\left(X\right)=\mathtt{WD}\left(X\right)$ hold for every $T_0$ space $X$? It is not difficult to check that the space $X$ is a WD space but not a Rudin space. Therefore, Example 4.15 in [22] also gave a negative answer to another related question raised by Xu et al. in [42]: Is every well-filtered determined space a Rudin space?

Proposition 4.7. ([36]) Suppose that $\mathbf{K}$ is adequate and closed with respect to homeomorphisms. Then a $T_0$ space $X$ is a $\mathbf{K}$-space if and only if $\mathbf{K}\left(X\right)={\mathcal{S}}_c\left(X\right)$.

Using Rudin sets and WD sets, one can obtain some characterizations of well-filtered spaces and sober spaces.

Proposition 4.8. ([42]) For a $T_0$ space $X$, the following conditions are equivalent:

1. $X$ is well-filtered.

2. $\mathtt{RD}\left(X\right)={\mathcal{S}}_c\left(X\right)$.

3. $\mathtt{WD}\left(X\right)={\mathcal{S}}_c\left(X\right)$.

Proposition 4.9. ([42]) For a $T_0$ space $X$, the following conditions are equivalent:

1. $X$ is sober.

2. $X$ is a DC $d$-space.

3. $X$ is a well-filtered DC space.

4. $X$ is a well-filtered Rudin space.

5. $X$ is a well-filtered WD space.

Lemma 4.10. ([54]) If $X$ is a locally hypercompact $T_0$ space and $A\in \mathrm{Irr}\left(X\right)$, then there is a directed subset $D\subseteq \downarrow A$ such that $\overline{A}=\overline{D}$ for. Therefore, $X$ is a DC space.

Theorem 4.11. ([42]) Let $X$ be a $T_0$ space.

1. If $X$ is locally compact, then it is a Rudin space.

2. If $X$ is core compact, then it is a WD space.

Figure 2 shows some relationships among some types of spaces.

Figure 2.

Certain relations among some kinds of spaces.

Question 4.12. ([42]) Is every core compact $T_0$ space a Rudin space?

Theorem 4.13. ([38]) Let $\mathbf{K}$ be a full subcategory of ${\mathbf{Top}}_d$ containing $\mathbf{Sob}$ and $X={\prod }_{i\in I}{X}_i$ the product of a family $\left\{X_i:i\in I\right\}$ of $T_0$ spaces. For $A\in \mathrm{Irr}\left(X\right)$, the following conditions are equivalent:

1. $A$ is a $\mathbf{K}$-determined set.

2. $p_i\left(A\right)$ is a $\mathbf{K}$-determined set for each $i\in I$.

From Lemma 2.5 and Theorem 4.13, we deduce the following.

Corollary 4.14. Let $\mathbf{K}$ be a full subcategory of ${\mathbf{Top}}_d$ containing $\mathbf{Sob}$ and $X={\prod }_{i\in I}{X}_i$ the product of a family $\left\{X_i:i\in I\right\}$ of $T_0$ spaces. If $A\in \mathbf{K}\left(X\right)$, then $A={\prod }_{i\in I}{p}_i\left(X_i\right)$, and $p_i\left(A\right)\in \mathbf{K}\left(X_i\right)$ for all $i\in I$.

Theorem 4.15. ([38]) Let $\mathbf{K}$ be a full subcategory of ${\mathbf{Top}}_d$ containing $\mathbf{Sob}$ and $X={\prod }_{i\in I}{X}_i$ the product of a family $\left\{X_i:i\in I\right\}$ of $T_0$ spaces. Then the following conditions are equivalent:

1. $X$ is a $\mathbf{K}$-determined space.

2. For each $i\in I$, $X_i$ is a $\mathbf{K}$-determined space.

Let $\mathbf{K}={\mathbf{Top}}_d$ or $\mathbf{K}={\mathbf{Top}}_w$. Then by Theorems 4.13 and 4.15, we deduce the following two corollaries.

Corollary 4.16. ([38]) Let $\left\{X_i:i\in I\right\}$ be a family of $T_0$ spaces and $X={\prod }_{i\in I}{X}_i$ the product space. For $A\in \mathrm{Irr}\left(X\right)$, the following conditions are equivalent:

1. $A$ is $d$-determined (resp., well-filtered determined).

2. $p_i\left(A\right)$ is $d$-determined (resp., well-filtered determined) for each $i\in I$.

Corollary 4.17. ([38]) For a family $\left\{X_i:i\in I\right\}$ of $T_0$ spaces, the following two conditions are equivalent:

1. The product space ${\prod }_{i\in I}{X}_i$ is a $d$-determined space (resp., a $\mathit{\text{mathsfWD}}$ space).

2. For each $i\in I$, $X_i$ is a $d$-determined space (resp., a $\mathit{\text{mathsfWD}}$ space).

5. K-reflections of ${\mathit{T}}_{\mathbf{0}}$ spaces

Definition 5.1. ([19, 36]) Let $\mathbf{K}$ be a full subcategory of ${\mathbf{Top}}_0$ containing $\mathbf{Sob}$ and $X$ a $T_0$ space. A $\mathbf{K}$-reflection of $X$ is a pair $⟨\tilde{X},\mu ⟩$ consisting of a $\mathbf{K}$-space $\tilde{X}$ and a continuous mapping $\mu :X\to \tilde{X}$ satisfying that for any continuous mapping $f:X\to Y$ to a $\mathbf{K}$-space, there exists a unique continuous mapping $f^{\ast }:\tilde{X}\to Y$ such that $f^{\ast }\circ \mu =f$, 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. The space of $\mathbf{Sob}$-reflection of $X$ is the sobrification $X^s$ of $X$. The space of ${\mathbf{Top}}_d$-reflection (resp., ${\mathbf{Top}}_w$-reflection) of $X$ is denoted by $X^d$ (resp., $X^w$).

It is well-known that $\mathbf{Sob}$ is reflective in ${\mathbf{Top}}_0$ (see [4, 7]). Using $d$-closures, Wyler [32] proved that ${\mathbf{Top}}_d$ is reflective in ${\mathbf{Top}}_0$ (see also [2, 3, 48]). In [19], it was proved by Wyler’s method that every Keimel-Lawson category $\mathbf{K}$ is reflective in ${\mathbf{Top}}_0$.

For quite a long time, it is not known whether ${\mathbf{Top}}_w$ is reflective in ${\mathbf{Top}}_0$. Recently, this problem was answered positively and independently in [22, 29, 31, 42]. In [36], for a full subcategory $\mathbf{K}$ of ${\mathbf{Top}}_0$ containing $\mathbf{Sob}$, a direct approach to $\mathbf{K}$-reflections of $T_0$ spaces were provided.

By Proposition 4.6, $\left\{{\diamond }_{\mathbf{K}\left(X\right)}U:U\in \mathcal{O}\left(X\right)\right\}$ is a topology on $\mathbf{K}\left(X\right)$. In the following, let ${\eta }_X^k:X\to P_H\left(\mathbf{K}\left(X\right)\right)$, ${\eta }_X^k\left(x\right)=\overline{\left\{x\right\}}$, be the canonical mapping from $X$ to $P_H\left(\mathbf{K}\left(X\right)\right)$. It is straightforward to verify that the canonical mapping ${\eta }_X^k:X\to P_H\left(\mathbf{K}\left(X\right)\right)$ is a topological embedding.

For the $\mathbf{K}$-reflections of $T_0$ spaces, the following lemma is crucial.

Lemma 5.2. ([36]) Let $X$ be a $T_0$ space and $f:X\to Y$ a continuous mapping from $X$ to a well-filtered space $Y$. Then there exists a unique continuous mapping $f^{\ast }:P_H\left(\mathbf{K}\left(X\right)\right)\to Y$ such that $f^{\ast }\circ {\eta }_X^k=f$, that is, the following diagram commutes.

Theorem 5.3. ([36]) Let $\mathbf{K}$ be a full subcategory of ${\mathbf{Top}}_0$ containing $\mathbf{Sob}$ and Let $X$ a $T_0$ space. If $P_H\left(\mathbf{K}\left(X\right)\right)$ is a $\mathbf{K}$-space, then the pair $⟨X^k=P_H\left(\mathbf{K}\left(X\right)\right),{\eta }_X^k⟩$, where ${\eta }_X^k:X\to X^k$, $x\mapsto \overline{\left\{x\right\}}$, is the $\mathbf{K}$-reflection of $X$.

For a full subcategory $\mathbf{K}$ of ${\mathbf{Top}}_0$ containing $\mathbf{Sob}$, $\mathbf{K}$ is called adequate if $P_H\left(\mathbf{K}\left(X\right)\right)$ is a $\mathbf{K}$-space for any $T_0$ space $X$.

By Theorem 5.3, we get the following.

Corollary 5.4. ([36]) If $\mathbf{K}$ is adequate, then $\mathbf{K}$ is reflective in ${\mathbf{Top}}_0$. For a $T_0$ space $X$, $X^k=P_H\left(\mathbf{K}\left(X\right)\right)$ with canonical mapping ${\eta }_X:X\to X^k$ is the $\mathbf{K}$-reflection of $X$.

Theorem 5.5. ([36]) If $\mathbf{K}$ is adequate, then for any $T_0$ spaces $X,Y$ and any continuous mapping $f:X\to Y$, there exists a unique continuous mapping $f^k:X^k\to Y^k$ such that $f^k\circ {\eta }_X^k={\eta }_Y^k\circ f$, that is, the following diagram commutes.

For each $A\in \mathbf{K}\left(X\right)$, $f^k\left(A\right)=\overline{f\left(A\right)}$. Therefore, a functor $K:{\mathbf{Top}}_0\to \mathbf{K}$ is defined, which is the left adjoint to the inclusion functor $I:\mathbf{K}\to {\mathbf{Top}}_0$.

Theorem 5.6. ([36]) If $\mathbf{K}\in \left\{\mathbf{Sob},{\mathbf{Top}}_d,{\mathbf{Top}}_w\right\}$ or $\mathbf{K}$ is a Keimel-Lawson category, then $\mathbf{K}$ is adequate and hence reflective in ${\mathbf{Top}}_0$. For a $T_0$ space $X$, $X^k=P_H\left(\mathbf{K}\left(X\right)\right)$ with the canonical mapping ${\eta }_X:X\to X^k$ is the $\mathbf{K}$-reflection of $X$.

Corollary 5.7. ([36]) ${\mathbf{Top}}_d$ is adequate and hence reflective in ${\mathbf{Top}}_0$. For a $T_0$ space $X$, $X^d=P_H\left(d\left(X\right)\right)$ with the canonical mapping ${\eta }_X:X\to X^d$ is the $d$-reflection of $X$.

Corollary 5.8. ([36]) ${\mathbf{Top}}_w$ is adequate and hence reflective in ${\mathbf{Top}}_0$. For a $T_0$ space $X$, $X^w=P_H\left(\mathtt{WD}\left(X\right)\right)$ with the canonical mapping ${\eta }_X:X\to X^w$ is the well-filtered reflection of $X$.

6. K-reflections of Scott spaces

In this section, some necessary and sufficient conditions are given for the $\mathbf{K}$-reflection of a $T_0$ space (especially, a Scott space) to be a Scott space.

Theorem 6.1. ([39]) Let $\mathbf{K}$ be a full subcategory of ${\mathbf{Top}}_d$ containing $\mathbf{Sob}$ which is adequate and closed with respect to homeomorphisms. For a $T_0$ space $X$, consider the following three conditions:

1. $\mathrm{\Sigma }\mathbf{K}\left(X\right)$ is a $\mathbf{K}$-space.

2. ${\eta }_X^{\sigma }:X\to \mathrm{\Sigma }\mathbf{K}\left(X\right)$, ${\eta }_X^{\sigma }\left(x\right)=\overline{\left\{x\right\}}$, is continuous.

3. The K-reflection $X^k$ of $X$ is a Scott space.

Then (1) + (2) $\Rightarrow$ (3), and(3) $\Rightarrow$ (1). Moreover, when Conditions (1) and (2) hold, the Scott space $\mathrm{\Sigma }\mathbf{K}\left(X\right)$ with the canonical mapping ${\eta }_X^{\sigma }:X\to \mathrm{\Sigma }\mathbf{K}\left(X\right)$, ${\eta }_X^{\sigma }\left(x\right)=\overline{\left\{x\right\}}$, is a $\mathbf{K}$-reflection of $X$.

In particular, we have the following result for the Scott space of a poset.

Theorem 6.2. ([39]) Let $\mathbf{K}$ be a full subcategory of ${\mathbf{Top}}_d$ containing $\mathbf{Sob}$ which is adequate and closed with respect to homeomorphisms. Then for any poset $P$, the following two conditions are equivalent:

1. $\mathrm{\Sigma }\mathbf{K}\left(\mathrm{\Sigma }P\right)$ is a K-space.

2. The K-reflection ${\left(\mathrm{\Sigma }P\right)}^k$ of $\mathrm{\Sigma }P$ is a Scott space.

Moreover, when Condition (1) holds, the Scott space $\mathrm{\Sigma }\mathbf{K}\left(\mathrm{\Sigma }P\right)$ with the canonical mapping ${\eta }_P^{\sigma }:\mathrm{\Sigma }P\to \mathrm{\Sigma }\mathbf{K}\left(\mathrm{\Sigma }P\right)$, ${\eta }_P^{\sigma }\left(x\right)={\mathrm{cl}}_{\sigma \left(P\right)}\left\{x\right\}=\downarrow x$, is a $\mathbf{K}$-reflection of $\mathrm{\Sigma }P$.

For the case of $\mathbf{K}={\mathbf{Top}}_d$, as an immediate corollary of Theorems 6.1 and 6.2), we get the following Corollary.

Corollary 6.3. ([19, 36]) Let $X$ be a $T_0$ space and $P$ a poset.

1. If ${\xi }_X:X\to \mathrm{\Sigma }d\left(X\right)$, ${\xi }_X\left(x\right)=\overline{\left\{x\right\}}$, is continuous, then the $d$-reflection $X^d$ of $X$ is the Scott space $\mathrm{\Sigma }d\left(X\right)$.

2. The poset $d\left(\mathrm{\Sigma }P\right)$ is a dcpo and the $d$-reflection ${\left(\mathrm{\Sigma }P\right)}^d$ of $\mathrm{\Sigma }P$ is the Scott space $\mathrm{\Sigma }d\left(\mathrm{\Sigma }P\right)$ with the canonical mapping ${\eta }_P:\mathrm{\Sigma }P\to \mathrm{\Sigma }d\left(X\right)$, given by ${\eta }_P\left(x\right)={\mathrm{cl}}_{\sigma \left(P\right)}\left\{x\right\}$ for each $x\in X$.

When $\mathbf{K}={\mathbf{Top}}_w$ in Theorems 6.1 and 6.2, we get the following two corollaries.

Corollary 6.4. ([39]) For a $T_0$ space $X$, consider the following three conditions:

1. $\mathrm{\Sigma }\mathrm{WD}\left(X\right)$ is a well-filtered space.

2. ${\eta }_X^w:X\to \mathrm{\Sigma }\mathtt{WD}\left(X\right)$, ${\eta }_X^w\left(x\right)=\overline{\left\{x\right\}}$, is continuous.

3. The well-filtered reflection $X^w$ of $X$ is a Scott space.

Then (1) + (2) $\Rightarrow$ (3), and (3) $\Rightarrow$ (1). Moreover, when Conditions (1) and (2) hold, the Scott space $\mathrm{\Sigma }\mathrm{WD}\left(X\right)$ with the canonical mapping ${\eta }_X^w:X\to \mathrm{\Sigma }\mathtt{WD}\left(X\right)$, ${\eta }_X^w\left(x\right)=\overline{\left\{x\right\}}$, is a well-filtered reflection of $X$.

Corollary 6.5. ([39]) For any poset $P$, the following two conditions are equivalent:

1. $\mathrm{\Sigma }\mathrm{WD}\left(\mathrm{\Sigma }P\right)$ is a well-filtered space.

2. The well-filtered reflection ${\left(\mathrm{\Sigma }P\right)}^w$ of $\mathrm{\Sigma }P$ is a Scott space.

Moreover, when Condition (1) holds, the Scott space $\mathrm{\Sigma }\mathrm{WD}\left(\mathrm{\Sigma }P\right)$ with the canonical mapping ${\eta }_P^w:\mathrm{\Sigma }P\to \mathrm{\Sigma }\mathrm{WD}\left(\mathrm{\Sigma }P\right)$, ${\eta }_P^w\left(x\right)={\mathrm{cl}}_{\sigma \left(P\right)}\left\{x\right\}=\downarrow x$, is a well-filtered reflection of $\mathrm{\Sigma }P$.

Theorem 6.6. ([39]) Let $\mathbf{K}$ be a full subcategory of ${\mathbf{Top}}_d$ containing $\mathbf{Sob}$ which is adequate and closed with respect to homeomorphisms. Suppose that $X$ is a $T_0$ space for which ${\mathrm{Irr}}_c\left(X\right)=\left\{\overline{\left\{x\right\}}:x\in X\right\}\cup \left\{X\right\}$ and $X$ is not a $\mathbf{K}$-space. Consider the following three conditions:

1. $\mathrm{\Sigma }{\left(\mathrm{\Omega }X\right)}_{{\rm T}}$ is a K-space.

2. ${\zeta }_X^{\sigma }:X\to \mathrm{\Sigma }{\left(\mathrm{\Omega }X\right)}_{{\rm T}}$, ${\zeta }_X^{\sigma }\left(x\right)=x$, is continuous.

3. The K-reflection $X^k$ of $X$ is a Scott space.

Then (1) + (2) $\Rightarrow$ (3), and (3) $\Rightarrow$ (1). Moreover, when Conditions (1) and (2) hold, the Scott space $\mathrm{\Sigma }{\left(\mathrm{\Omega }X\right)}_{{\rm T}}$ with the canonical mapping ${\zeta }_X^{\sigma }:X\to \mathrm{\Sigma }{\left(\mathrm{\Omega }X\right)}_{{\rm T}}$, ${\zeta }_X^{\sigma }\left(x\right)=x$, is a K-reflection of $X$.

Definition 6.7. ([39]) For a poset $P$, ∂ let $P_{{\rm T}}=P\cup \left\{\mathrm{T}\right\}$ ($\mathrm{T}\notin P$) denote the poset obtained from $P$ by adjoining the largest element $\mathrm{T}$ (whether $P$ has one or not).

Clearly, the order on $P_{\mathrm{T}}$ is as follows: $x\le y$ iff $x\le y$ in $P$ or $y=\mathrm{T}$. The element $\mathrm{T}$ is the largest element of $P_{{\rm T}}$ (even $P$ has the largest element).

It is straightforward to verify the following result.

Lemma 6.8. ([39]) Let $P$ be a dcpo. Then

1. $P_{\mathrm{T}}$ is a dcpo and in $P_{\mathrm{T}}$ we have $\mathrm{T}\ll \mathrm{T}$, i.e., $\uparrow \mathrm{T}=\left\{\mathrm{T}\right\}\in \sigma \left(P_{\mathrm{T}}\right)$,

2. ${\zeta }_P:\mathrm{\Sigma }P\to \mathrm{\Sigma }{P}_{\mathrm{T}}$, $x\mapsto x$, is continuous.

By Lemma 6.8 and Theorem 6.6, we get the following.

Corollary 6.9. ([39]) Let $\mathbf{K}$ be a full subcategory of ${\mathbf{Top}}_d$ containing $\mathbf{Sob}$ which is adequate and closed with respect to homeomorphisms. Suppose that $P$ is a poset for which ${\mathrm{Irr}}_c\left(\mathrm{\Sigma }P\right)=\left\{\overline{\left\{x\right\}}:x\in P\right\}\cup \left\{P\right\}$ and $\mathrm{\Sigma }P$ is not a $\mathbf{K}$-space. Then the following two conditions are equivalent:

1. $\mathrm{\Sigma }{P}_{\mathrm{T}}$ is a K-space.

2. The K-reflection ${\left(\mathrm{\Sigma }P\right)}^k$ of $\mathrm{\Sigma }P$ is a Scott space.

Moreover, when Condition (1) holds, the Scott space $\mathrm{\Sigma }{P}_{\mathrm{T}}$ with the embedding $i_P:\mathrm{\Sigma }P\to \mathrm{\Sigma }{P}_{\mathrm{T}}$, $i_P\left(x\right)=x$, is a K-reflection of $X$.

Now we give some examples and counterexamples related to the $\mathbf{K}$-reflections of $T_0$ spaces (esp., Scott spaces).

Example 6.10. ([39]) Let $\mathbf{K}$ be a full subcategory of ${\mathbf{Top}}_d$ containing $\mathbf{Sob}$ which is adequate and closed with respect to homeomorphisms. Since $\mathbb{N}$ is not a dcpo, $\mathrm{\Sigma }\mathbb{N}$ is not a $d$-space and hence not a K-space. Clearly, ${\mathrm{Irr}}_c\left(\mathrm{\Sigma }\mathbb{N}\right)=\left\{\overline{\left\{n\right\}}=\downarrow n:n\in \mathbb{N}\right\}\cup \left\{\mathbb{N}\right\}$. As ${\mathbb{N}}_{{\rm T}}$ is an algebraic lattice, by ([4], Proposition III-3.7), $\mathrm{\Sigma }{\mathbb{N}}_{{\rm T}}$ is a sober space and hence a K-space. By Corollary 6.9, the K-reflection of $\mathrm{\Sigma }\mathbb{N}$ is a Scott space. More precisely, $\mathrm{\Sigma }{\mathbb{N}}_{\mathrm{T}}$ with the embedding $i_{\mathbb{N}}:\mathrm{\Sigma }\mathbb{N}\to \mathrm{\Sigma }{\mathbb{N}}_{\mathrm{T}}$, $i_{\mathbb{N}}\left(n\right)=n$, is a K-reflection of $\mathrm{\Sigma }\mathbb{N}$.

Example 6.11. ([39]) Let $\mathbf{K}$ be a full subcategory of ${\mathbf{Top}}_d$ containing $\mathbf{Sob}$ which is adequate and closed with respect to homeomorphisms and $P=\mathbb{N}\cup \left\{a,b\right\}$. Define a partial order $\le$ on $P$ as follows:

1. $n<n+1$ for each $n\in \mathbb{N}$,

2. $n<a$ and $n<b$ for all $n\in \mathbb{N}$, and

3. $a$ and $b$ are incomparable.

Then $max\left(P\right)=\left\{a,b\right\}$ and $P$ is not a dcpo since the chain $\mathbb{N}$ does not have a least upper bound in $P$. So $\mathrm{\Sigma }P$ is not a $d$-space and hence not a K-space. It is easy to verify that ${\mathrm{Irr}}_c\left(\mathrm{\Sigma }P\right)=\left\{\downarrow x:x\in P\right\}\cup \left\{\mathbb{N}\right\}$. Hence by Propositions 4.6 and 4.7, $\mathbf{K}\left(\mathrm{\Sigma }P\right)={\mathrm{Irr}}_c\left(\mathrm{\Sigma }P\right)=\left\{\downarrow x:x\in P\right\}\cup \left\{\mathbb{N}\right\}$. Now we show that $\mathrm{\Sigma }\mathbf{K}\left(\mathrm{\Sigma }P\right)$ is sober. Let $Q=\mathbb{N}\cup \left\{a,b,c\right\}$. Define a partial order ${\le }_Q$ on $Q$ as follows:

1. for $x,y\in P$, $x{\le }_{Q}y$ iff $x{\le }_{P}y$ in $P$,

2. $n{<}_{Q}c$ for all $n\in \mathbb{N}$, and

3. $c{<}_{Q}a$ and $c{<}_{Q}b$.

Clearly, $Q$ is an algebraic domain and $K\left(Q\right)=\mathbb{N}\cup \left\{a,b\right\}$. Define a mapping $\psi :\mathbf{K}\left(\mathrm{\Sigma }P\right)\to Q$ by

$$\psi \left(x\right)=\left\{\begin{array}{ll}n,& x=\uparrow n\left(n\in \mathbb{N}\right),\\ c,& x=\mathbb{N},\\ a,& x=\left\{a\right\},\\ b,& x=\left\{b\right\}.\end{array}\right.$$

It is straightforward to verify that $\psi$ is a poset isomorphism, and hence induces a homeomorphism from $\mathrm{\Sigma }\mathbf{K}\left(\mathrm{\Sigma }P\right)$ to $\mathrm{\Sigma }Q$. Clearly, $Q$ is a dcpo, $K\left(Q\right)=\mathbb{N}\cup \left\{a,b\right\}$ and $c={\vee }_Q\mathbb{N}$, whence $Q$ is an algebraic domain. By ([4], Proposition III-3.7), $\mathrm{\Sigma }Q$ is sober, and consequently, $\mathrm{\Sigma }\mathbf{K}\left(\mathrm{\Sigma }P\right)$ is a sober space and hence a K-space. It follows from Theorem 6.2 that the K-reflection of $\mathrm{\Sigma }P$ is a Scott space. More precisely, $\mathrm{\Sigma }Q$ with the embedding $i_P=\psi \circ {\eta }_P^{\sigma }:\mathrm{\Sigma }P\to \mathrm{\Sigma }Q$, $i_P\left(x\right)=x$, is a K-reflection of $\mathrm{\Sigma }P$.

The following two examples show that for a $T_0$ space $X$, Condition (1) of Theorem 6.1 is only a necessary condition but not a sufficient condition for the K-reflection of $X$ to be a Scott space.

Example 6.12. Let $\mathbf{K}$ be a full subcategory of ${\mathbf{Top}}_w$ containing $\mathbf{Sob}$ which is adequate and closed with respect to homeomorphisms. Let $X$ be a countably infinite set and endow $X$ with the co-finite topology (having the finite sets as closed sets). The resulting space is denoted by $X_{\mathit{cof}}$. Then

1. $\mathcal{C}\left(X_{\mathit{cof}}\right)=\left\{\varnothing ,X\right\}\cup X^{\left(<\omega \right)}$, $X_{\mathit{cof}}$ is $T_1$ and hence a $d$-space.

2. $\mathtt{K}\left(X_{\mathit{cof}}\right)=2^X\backslash \left\{\varnothing \right\}$.

3. $X_{\mathit{cof}}$ is locally compact and first-countable.

4. $X_{\mathit{cof}}$ is not well-filtered and hence not a K-space.

   Let $\mathcal{K}=\left\{X\backslash F:F\in X^{\left(<\omega \right)}\right\}$. Then $\mathcal{K}$ is a filtered family of saturated compact subsets of $X_{\mathit{cof}}$ and $\cap \mathcal{K}=\varnothing$, but $X\backslash F\ne \varnothing$ for every $F\in X^{\left(<\omega \right)}$. Thus $X_{\mathit{cof}}$ is not well-filtered.

5. $\mathbf{K}\left(X_{\mathit{cof}}\right)={\mathrm{Irr}}_c\left(X_{\mathit{cof}}\right)=\left\{\left\{x\right\}:x\in X\right\}\cup \left\{X\right\}$.

   It is easy to see that ${\mathrm{Irr}}_c\left(X_{\mathit{cof}}\right)=\left\{\left\{x\right\}:x\in X\right\}\cup \left\{X\right\}$. By Propositions 4.6 and 4.7, $\mathbf{K}\left(X_{\mathit{cof}}\right)={\mathrm{Irr}}_c\left(X_{\mathit{cof}}\right)=\left\{\left\{x\right\}:x\in X\right\}\cup \left\{X\right\}$.

6. $\mathrm{\Sigma }\mathbf{K}\left(X_{\mathit{cof}}\right)$ is sober and hence a K-space.

   Clearly, $\mathbf{K}\left(X_{\mathit{cof}}\right)$ (with the order of set inclusion) is a Noetherian dcpo, that is, it satisfies the ascending chain condition: every ascending chain has a greatest member. Whence $\mathbf{K}\left(X_{\mathit{cof}}\right)$ is an algebraic domain. By ([4], Proposition III-3.7), $\mathrm{\Sigma }\mathbf{K}\left(X_{\mathit{coc}}\right)$ is sober, whence it is a K-space.

7. ${\eta }_{X_{\mathit{cof}}}^{\sigma }:X_{\mathit{cof}}\to \mathrm{\Sigma }{\mathrm{Irr}}_c\left(X_{\mathit{cof}}\right)$, $x\mapsto \left\{x\right\}$, is not continuous.

   Let $C\notin \mathcal{O}\left(X_{\mathit{cof}}\right)$. Then $\left\{\left\{x\right\}:x\in C\right\}\cup \left\{X\right\}\in \sigma \left({\mathrm{Irr}}_c\left(X_{\mathit{cof}}\right)\right)$, but ${\left({\eta }_{X_{\mathit{cof}}}^{\sigma }\right)}^{-1}\left(\left\{\left\{x\right\}:x\in C\right\}\cup \left\{X\right\}\right)=C\notin \mathcal{O}\left(X_{\mathit{cof}}\right)$, proving that ${\eta }_{X_{\mathit{cof}}}^{\sigma }:X_{\mathit{cof}}\to \mathrm{\Sigma }{\mathrm{Irr}}_c\left(X_{\mathit{cof}}\right)$ is not continuous.

8. The K-reflection of $X_{\mathit{cof}}$ is not a Scott space. In particular, the well-filtered reflection of $X_{\mathit{cof}}$ is not a Scott space and the sobrification of $X_{\mathit{cof}}$ is also not a Scott space.

Assume, on the contrary, that the K-reflection of $X_{\mathit{cof}}$ is a Scott space. Then there is a poset $P$ such that ${\left(X_{\mathit{cof}}\right)}^k=P_H\left(\mathbf{K}\left(X_{\mathit{cof}}\right)\right)$ is homeomorphic to $\mathrm{\Sigma }P$, whence by (e), ${\mathrm{Irr}}_c\left(X_{\mathit{cof}}\right)=\mathbf{K}\left(X_{\mathit{cof}}\right)\left(=\mathrm{\Omega }{P}_H\left({\mathrm{Irr}}_c\left(X_{\mathit{cof}}\right)\right)\right)$ and $P\left(=\mathrm{\Omega \Sigma }P\right)$ are isomorphic. It follows that $\mathrm{\Sigma }{\mathrm{Irr}}_c\left(X_{\mathit{cof}}\right)$ and $\mathrm{\Sigma }P$ are homeomorphic, and consequently, $P_H\left({\mathrm{Irr}}_c\left(X_{\mathit{cof}}\right)\right)\cong \mathrm{\Sigma }{\mathrm{Irr}}_c\left(X_{\mathit{cof}}\right)$. Therefore, $\mathcal{O}\left(P_H\left({\mathrm{Irr}}_c\left(X_{\mathit{cof}}\right)\right)\right)\cong \sigma \left({\mathrm{Irr}}_c\left(X_{\mathit{cof}}\right)\right)$, and hence $2^{\omega }=\mid \sigma \left({\mathrm{Irr}}_c\left(X_{\mathit{cof}}\right)\right)\mid =\mid \mathcal{O}\left(P_H\left({\mathrm{Irr}}_c\left(X_{\mathit{cof}}\right)\right)\right)\mid \le \mid \mathcal{O}\left(X_{\mathit{cof}}\right)\mid =\mid X^{\left(<\omega \right)}\mid =\omega$, which is a contradiction by Cantor’s Theorem (see [57], III-2.13 Cantor’s Theorem). So the K-reflection of $X_{\mathit{cof}}$ is not a Scott space.

Example 6.13. ([39]) Let $X=2^{\mathbb{N}}$ (the set of all subsets of $\mathbb{N}$) and endow $X$ with the co-countable topology (having the countable sets as closed sets). The resulting space is denoted by $X_{\mathit{coc}}$. Then

1. $\mid X\mid =\mathfrak{c}=2^{\omega }$ (where $\mathfrak{c}=\mid \mathrm{ℝ}\mid$ and $\mathrm{ℝ}$ is the set of all reals) and $X$ is an uncountably infinite set.

2. $X_{\mathit{coc}}$ is $T_1$ and $\mathcal{C}\left(X_{\mathit{coc}}\right)=\left\{\varnothing ,X\right\}\cup X^{\left(⩽\omega \right)}$.

3. $\mathtt{K}\left(X_{\mathit{coc}}\right)=X^{\left(<\omega \right)}\backslash \left\{\varnothing \right\}$.

4. $X_{\mathit{coc}}$ is well-filtered.

5. ${\mathrm{Irr}}_c\left(X_{\mathit{coc}}\right)=\left\{\left\{x\right\}:x\in X\right\}\cup \left\{X\right\}$.

6. $\mathrm{\Sigma }{\mathrm{Irr}}_c\left(X_{\mathit{coc}}\right)$ is sober.

   Let $P=\left\{\left\{x\right\}:x\in X\right\}\cup \left\{X\right\}$ with the order of set inclusion. It is easy to see that $P$ is a Notherian dcpo and hence $\mathrm{\Sigma }P$ is sober by ([4], Proposition III-3.7). Clearly, $\sigma \left(P\right)=\gamma \left(P\right)$ and hence $\mid \sigma \left(P\right)\mid =\mid \gamma \left(P\right)\mid =2^{\mathfrak{c}}$.

7. ${\eta }_{X_{\mathit{coc}}}^{\sigma }:X_{\mathit{coc}}\to \mathrm{\Sigma }{\mathrm{Irr}}_c\left(X_{\mathit{coc}}\right)$, $x\mapsto \left\{x\right\}$, is not continuous.

   Let $C$ be any non-countable proper subset of $X$, that is, $C\notin \mathcal{C}\left(X_{\mathit{coc}}\right)$. Then $\left\{\left\{x\right\}:x\in C\right\}\in \mathcal{C}\left(\mathrm{\Sigma }\left({\mathrm{Irr}}_c\left(X_{\mathit{coc}}\right)\right)\right)$, but ${\left({\eta }_{X_{\mathit{coc}}}^{\sigma }\right)}^{-1}\left(\left\{\left\{x\right\}:x\in C\right\}\right)=C\notin \mathcal{C}\left(X_{\mathit{coc}}\right)$, proving that ${\eta }_{X_{\mathit{coc}}}^{\sigma }:X_{\mathit{coc}}\to \mathrm{\Sigma }{\mathrm{Irr}}_c\left(X_{\mathit{coc}}\right)$ is not continuous.

8. The sobrification of $X_{\mathit{coc}}$ is not a Scott space.

   Assume, on the contrary, that the sobrification of $X_{\mathit{coc}}$ is a Scott space. Then there is a poset $P$ such that ${\left(X_{\mathit{coc}}\right)}^s=P_H\left({\mathrm{Irr}}_c\left(X_{\mathit{coc}}\right)\right)$ is homeomorphic to $\mathrm{\Sigma }P$, whence ${\mathrm{Irr}}_c\left(X_{\mathit{coc}}\right)\left(=\mathrm{\Omega }{P}_H\left({\mathrm{Irr}}_c\left(X_{\mathit{coc}}\right)\right)\right)$ and $P\left(=\mathrm{\Omega \Sigma }P\right)$ are isomorphic. It follows that $\mathrm{\Sigma }{\mathrm{Irr}}_c\left(X_{\mathit{coc}}\right)$ and $\mathrm{\Sigma }P$ are homeomorphic, and consequently, $P_H\left({\mathrm{Irr}}_c\left(X_{\mathit{coc}}\right)\right)\cong \mathrm{\Sigma }{\mathrm{Irr}}_c\left(X_{\mathit{coc}}\right)$. Therefore, $\mathcal{O}\left(P_H\left({\mathrm{Irr}}_c\left(X_{\mathit{coc}}\right)\right)\right)\cong \sigma \left({\mathrm{Irr}}_c\left(X_{\mathit{coc}}\right)\right)$, and hence $2^{\mathfrak{c}}=\mid \sigma \left({\mathrm{Irr}}_c\left(X_{\mathit{coc}}\right)\right)\mid =\mid \mathcal{O}\left(P_H\left({\mathrm{Irr}}_c\left(X_{\mathit{coc}}\right)\right)\right)\mid \le \mid \mathcal{O}\left(X_{\mathit{coc}}\right)\mid ={\left(2^{\omega }\right)}^{\omega }=2^{\omega \cdot \omega }=2^{\omega }=\mathfrak{c}$ (see [57], III-3.23 Corollary and III-3.29 Proposition), which is a contradiction by Cantor’s Theorem. So the sobrification of $X_{\mathit{coc}}$ is not a Scott space.

Proposition 6.14. ([39]) Let $\mathbf{K}$ be a full subcategory of ${\mathbf{Top}}_w$ containing $\mathbf{Sob}$ which is adequate and closed with respect to homeomorphisms. Then the K-reflection of Johnstone’s space $\mathrm{\Sigma }\mathbb{J}$ is not a Scott space. In particular, neither the sobrification nor the well-filtered reflection of $\mathrm{\Sigma }\mathbb{J}$ is a Scott space.

7. K-reflections of product spaces

Theorem 7.1. ([38]) Let $\mathbf{K}$ be a full subcategory of ${\mathbf{Top}}_d$ containing $\mathbf{Sob}$. If $\mathbf{K}$ is adequate, then for any family $\left\{X_i:i\in I\right\}$ of $T_0$ spaces, ${\left({\prod }_{i\in I}{X}_i\right)}^k={\prod }_{i\in I}{X}_i^k$ (up to homeomorphism).

Corollary 7.2. ([11, 12]) For any family $\left\{X_i:i\in I\right\}$ of $T_0$ spaces, ${\left({\prod }_{i\in I}{X}_i\right)}^s={\prod }_{i\in I}{X}_i^s$ (up to homeomorphism).

By Proposition 4.6, $\mathbf{Sob}$ is adequate, and hence by Theorem 7.1, we have the following result.

Corollary 7.3. ([11, 12]) For a family $\left\{X_i:i\in I\right\}$ of $T_0$ spaces, ${\left({\prod }_{i\in I}{X}_i\right)}^s={\prod }_{i\in I}{X}_i^s$ (up to homeomorphism).

By Corollary 5.7 and Theorem 7.1, we have the following corollary.

Corollary 7.4. ([38]) For a family $\left\{X_i:i\in I\right\}$ of $T_0$ spaces, ${\left({\prod }_{i\in I}{X}_i\right)}^d={\prod }_{i\in I}{X}_i^d$ (up to homeomorphism).

Proposition 7.5. ([38]) Let $\left\{P_i:i\in I\right\}$ be a family of posets. If the Scott topology of the product ${\prod }_{i\in I}{P}_i$ is the product of the Scott topologies of the factors, then the $d$-reflection ${\left({\prod }_{i\in I}\mathrm{\Sigma }{P}_i\right)}^d$ of ${\prod }_{i\in I}\mathrm{\Sigma }{P}_i$ is the Scott space $\mathrm{\Sigma }d\left(\mathrm{\Sigma }{\prod }_{i\in I}{P}_i\right))$.

By Corollary 5.8 and Theorem 7.1, we deduce the following result.

Corollary 7.6. ([38]) For a family $\left\{X_i:i\in I\right\}$ of $T_0$ spaces, ${\left({\prod }_{i\in I}{X}_i\right)}^w={\prod }_{i\in I}{X}_i^w$ (up to homeomorphism).

By Theorems 5.6 and 7.1, we have the following corollary.

Corollary 7.7. ([38]) Let $\mathbf{K}$ be a Keimel-Lawson category. If $\mathbf{K}$ is a full subcategory of ${\mathbf{Top}}_d$, then for any family $\left\{X_i:i\in I\right\}$ of $T_0$ spaces, ${\left({\prod }_{i\in I}{X}_i\right)}^k={\prod }_{i\in I}{X}_i^k$ (up to homeomorphism).

Proposition 7.8. ([38]) Let $H:{\mathbf{Top}}_0\to \mathbf{Set}$ be an R-subset system and $X={\prod }_{i\in I}{X}_i$ the product space of a family $\left\{X_i:i\in I\right\}$ of $T_0$ spaces. For $A\in \mathrm{Irr}\left(X\right)$, the following two conditions are equivalent:

1. $A$ is H-sober determined (resp., super H-sober determined).

2. $p_i\left(A\right)$ is H-sober determined (resp., super H-sober determined) for each $i\in I$.

Proposition 7.9. Let $H:{\mathbf{Top}}_0\to \mathbf{Set}$ be an R-subset system and $\left\{X_i:i\in I\right\}$ a family of $T_0$ spaces. Then ${\left({\prod }_{i\in I}{X}_i\right)}^h={\prod }_{i\in I}{X}_i^h$ (up to homeomorphism).

Proposition 7.10. Let $H:{\mathbf{Top}}_0\to \mathbf{Set}$ be an R-subset system having property M and $\left\{X_i:i\in I\right\}$ a family of $T_0$ spaces. Then ${\left({\prod }_{i\in I}{X}_i\right)}^H={\prod }_{i\in I}{X}_i^H$ (up to homeomorphism).

For the concepts and notations used in the above three propositions, please see [41].

From Theorem 7.1, we deduce the following.

Corollary 7.11. ([36]) Suppose that $\mathbf{K}$ is adequate and closed with respect to homeomorphisms. Then for any family $\left\{X_i:i\in I\right\}$ of $T_0$ spaces, the following two conditions are equivalent:

1. The product space ${\prod }_{i\in I}{X}_i$ is a K-space.

2. For each $i\in I$, $X_i$ is a K-space.

By Theorem 5.6 and Corollary 7.11, we get the following corollary.

Corollary 7.12. If $\mathbf{K}\in \left\{\mathbf{Sob},{\mathbf{Top}}_d,{\mathbf{Top}}_w\right\}$ or $\mathbf{K}$ is a Keimel-Lawson category, then for any family $\left\{X_i:i\in I\right\}$ of $T_0$ spaces, the following two conditions are equivalent:

1. The product space ${\prod }_{i\in I}{X}_i$ is a K-space.

2. For each $i\in I$, $X_i$ is a K-space.

In particular, we have the following.

Corollary 7.13. ([29, 42, 44]) For any family $\left\{X_i:i\in I\right\}$ of $T_0$ spaces, the following two conditions are equivalent:

1. The product space ${\prod }_{i\in I}{X}_i$ is well-filtered.

2. For each $i\in I$, $X_i$ is well-filtered.

8. ${\mathbf{\text{DCPO}}}_{\mathit{s}}$-completions of posets

Let $\mathbf{\text{Poset}}$ denote the category of all posets with monotone (i.e. order-preserving) mappings, ${\mathbf{\text{DCPO}}}_s$ the category of all dcpos with Scott continuous mappings, and ${\mathbf{\text{Poset}}}_s$ the category of all posets with Scott continuous mappings. Then ${\mathbf{\text{DCPO}}}_s$ is a full subcategory of ${\mathbf{\text{Poset}}}_s$.

Definition 8.1. ([50]) A ${\mathbf{\text{DCPO}}}_s$-completion of a poset $P$, ${\mathbf{D}}_s$-completion of $P$ for short, is a pair $⟨\tilde{P},\eta ⟩$ consisting of a dcpo $\tilde{P}$ and a Scott continuous mapping $\eta :P\to \tilde{P}$, such that for any Scott continuous mapping $f:P\to Q$ to a dcpo $Q$, there exists a unique Scott continuous mapping $\tilde{f}:\tilde{P}\to Q$ such that $\tilde{f}\circ \eta =f$, that is, the following diagram commutes.

${\mathbf{D}}_s$-completions, if they exist, are unique up to isomorphism. We shall use ${\mathbf{D}}_s\left(P\right)$ to denote the ${\mathbf{D}}_{\mathbf{s}}$-completion of $P$ if it exists.

In [50], using the $D$-topologies and $d$-closure defined in [50] (see also [19]), which originates from Wyler [32], Zhao and Fan proved that for any poset $P$, the ${\mathbf{D}}_s$-completion of $P$ exists. As Keimel and Lawson pointed out in [19] that the ${\mathbf{D}}_s$-completion of a poset $P$ is a special case of the $d$-reflection of a certain $T_0$ space. More precisely, the $d$-reflection of Scott space $\mathrm{\Sigma }P$.

Proposition 8.2. ([36]) For a poset $P$, ${\mathbf{D}}_s\left(P\right)=d\left(\mathrm{\Sigma }P\right)$ with the canonical mapping ${\eta }_P:P\to {\mathbf{D}}_s\left(P\right)$, ${\eta }_P\left(x\right)={\mathrm{cl}}_{\sigma \left(P\right)}\left\{x\right\}$, is the ${\mathbf{D}}_s$-completion of $P$.

Corollary 8.3. ([36, 50]) ${\mathbf{\text{DCPO}}}_s$ is reflective in ${\mathbf{\text{Poset}}}_s$.

Remark 8.4. ([39]) The ${\mathbf{\text{DCPO}}}_s$-completion (that is, ${\mathbf{\text{DCPO}}}_s$-completion) was called the $\mathbf{D}$-completion in ([50], Definition 1) and the $\mathbf{\text{DCPO}}$-completion in [36]. For the sake of distinction, here we call such a completion the ${\mathbf{\text{DCPO}}}_s$-completion and give the $\mathbf{\text{DCPO}}$-completion (or $\mathbf{D}$-completion) a little different meaning (see Section 9).

By Propositions 7.5 and 8.2, we get the following corollary, which generalizes ([19], Proposition 7.8).

Proposition 8.5. ([38]) For a family $\left\{P_i:i\in I\right\}$ of posets, the product ${\prod }_{i\in I}{\mathbf{D}}_s\left(P_i\right)$ is the ${\mathbf{D}}_s$-completion of the product ${\prod }_{i\in I}{P}_i$, provided the Scott topology of the product ${\prod }_{i\in I}{P}_i$ is the product of the Scott topologies of the factors.

9. Scott $\mathbf{K}$-completions of posets

Definition 9.1. ([39]) Let $\mathbf{K}$ be a full subcategory of ${\mathbf{Top}}_d$ containing $\mathbf{Sob}$. A poset $P$ is called a Scott K-dcpo, a K-dcpo for short, if $\mathrm{\Sigma }P$ is a K-space. A poset (even a dcpo) $Q$ is said to be a non-$\mathbf{K}$ dcpo if $Q$ is not a K-dcpo. Let $\mathbf{K}$-${\mathbf{\text{DCPO}}}_s$ denote the category of all K-dcpos with Scott continuous mappings.

$\mathbf{K}‐{\mathbf{\text{DCPO}}}_s$ is a full subcategory of ${\mathbf{\text{DCPO}}}_s$, and it is a subcategory of $\mathbf{\text{DCPO}}$, but not a full subcategory of $\mathbf{\text{DCPO}}$.

Clearly, a poset $P$ is a ${\mathbf{Top}}_d$-dcpo ($\mathbf{d}$-dcpo for short) iff $P$ is a dcpo. For $\mathbf{K}={\mathbf{Top}}_w$, the K-dcpos are simply called the $\mathbf{WF}$-dcpos, and the category ${\mathbf{Top}}_w$-${\mathbf{\text{DCPO}}}_s$ is simply denoted as $\mathbf{WF}$-${\mathbf{\text{DCPO}}}_s$.

Definition 9.2. ([39]) Let $\mathbf{K}$ be a full subcategory of ${\mathbf{Top}}_d$ containing $\mathbf{Sob}$. A Scott K-completion, ${\mathbf{K}}_s$-completion for short, of a poset $P$ is a pair $⟨\tilde{P},\eta ⟩$ consisting of a $\mathbf{K}$-dcpo $\tilde{P}$ and a Scott continuous mapping $\eta :P\to \tilde{P}$, such that for any Scott continuous mapping $f:P\to Q$ to a K-dcpo $Q$, there exists a unique Scott continuous mapping $\tilde{f}:\tilde{P}\to Q$ such that $\tilde{f}\circ \eta =f$, that is, the following diagram commutes.

When $\mathbf{K}={\mathbf{Top}}_d$, the ${\mathbf{K}}_s$-completion is the ${\mathbf{\text{DCPO}}}_s$-completion. For $\mathbf{K}={\mathbf{Top}}_w$, the ${\mathbf{K}}_s$-completion is simply called the ${\mathbf{WF}}_s$-completion.

By a standard argument, ${\mathbf{K}}_s$-completions, if they exist, are unique up to isomorphism. We use ${\mathbf{K}}_s\left(P\right)$ to denote the ${\mathbf{K}}_s$-completion of $P$ if it exists. We will use ${\mathbf{D}}_s\left(P\right)$, ${\mathbf{WF}}_s\left(P\right)$ and ${\mathbf{Sob}}_s\left(P\right)$ to denote the ${\mathbf{\text{DCPO}}}_s$-completion, ${\mathbf{WF}}_s$-completion, and ${\mathbf{Sob}}_s$-completion of $P$, respectively.

Definition 9.3. ([39]) Let $\mathbf{K}$ be a full subcategory of ${\mathbf{Top}}_d$ containing $\mathbf{Sob}$. A $\mathbf{K}$-completion of a poset $P$ is a pair $⟨\tilde{P},\varphi ⟩$ consisting of a $\mathbf{K}$-dcpo $\tilde{P}$ and a monotone mapping $\varphi :P\to \tilde{P}$, such that for any monotone mapping $f:P\to Q$ to a $\mathbf{K}$-dcpo $Q$, there exists a unique Scott continuous mapping $\tilde{f}:\tilde{P}\to Q$ such that $\tilde{f}\circ \varphi =f$.

When $\mathbf{K}={\mathbf{Top}}_d$ the $\mathbf{K}$-completion is the $\mathbf{\text{DCPO}}$-completion. For $\mathbf{K}={\mathbf{Top}}_w$), the $\mathbf{K}$-completion is simply called the $\mathbf{WF}$-completion.

Similarly, $\mathbf{K}$-completions, if they exist, are unique up to isomorphism. We use $\mathbf{K}\left(P\right)$ to denote the $\mathbf{K}$-completion of $P$ if it exists. We will use $\mathbf{D}\left(P\right)$, $\mathbf{WF}\left(P\right)$ and $\mathbf{Sob}\left(P\right)$ to denote the $\mathbf{\text{DCPO}}$-completion of $P$, $\mathbf{WF}$-completion, and $\mathbf{Sob}$-completion of $P$, respectively.

Definition 9.4. ([39]) Let $\mathbf{K}$ be a full subcategory of ${\mathbf{Top}}_d$ containing $\mathbf{Sob}$. A poset $P$ is called a weak K-dcpo if there is a $\mathbf{K}$-space such that $P$ is isomorphic to $\mathrm{\Omega }X$.

Clearly, every weak K-dcpo is a dcpo, and a poset $P$ is a dcpo iff $P$ is a d-dcpo iff $P$ is a weak d-dcpo. By Corollary 3.3, every complete lattice is a WF-dcpo and is also a weak $\mathbf{Sob}$-dcpo. The Isbell’s lattice $L$, as a complete lattice, is a weak $\mathbf{Sob}$-dcpo but not a $\mathbf{Sob}$-dcpo.

Theorem 9.5. ([39]) Let $\mathbf{K}$ be a full subcategory of ${\mathbf{Top}}_d$ containing $\mathbf{Sob}$ which is adequate and closed with respect to homeomorphisms. For a poset $P$, if $\mathbf{K}\left(\mathrm{\Sigma }P\right)$ is a K-dcpo, then ${\mathbf{K}}_s\left(P\right)=\mathbf{K}\left(\mathrm{\Sigma }P\right)$ with the canonical mapping ${\eta }_P:P\to {\mathbf{K}}_s\left(P\right)$, ${\eta }_P\left(x\right)={\mathrm{cl}}_{\sigma \left(P\right)}\left\{x\right\}=\downarrow x$, is a ${\mathbf{K}}_s$-completion of $P$.

Definition 9.6. ([39]) Let $\mathbf{K}$ be a full subcategory of ${\mathbf{Top}}_0$. A poset $P$ is called a $S_{\mathbf{K}}$-poset if $\mathrm{\Sigma }\mathbf{K}\left(\mathrm{\Sigma }P\right)$ is a $\mathbf{K}$-space. Let ${\mathbf{S}}_{\mathbf{K}}$-${\mathbf{\text{Poset}}}_s$ denote the category of all $S_{\mathbf{K}}$-posets with Scott continuous mappings.

Clearly, ${\mathbf{S}}_{\mathbf{K}}$-${\mathbf{\text{Poset}}}_s$ is a full subcategory of ${\mathbf{\text{Poset}}}_s$. If $\mathbf{K}$ is a full subcategory of ${\mathbf{Top}}_d$ containing $\mathbf{Sob}$, then by Proposition 4.7, every $\mathbf{K}$-dcpo is a $S_{\mathbf{K}}$-poset, and hence $\mathbf{K}$-${\mathbf{\text{DCPO}}}_s$ is a full subcategory of ${\mathbf{S}}_{\mathbf{K}}$-${\mathbf{\text{Poset}}}_s$.

From Theorem 9.5 we deduce the following result.

Corollary 9.7. ([39]) Let $\mathbf{K}$ be a full subcategory of ${\mathbf{Top}}_d$ containing $\mathbf{Sob}$ which is adequate and closed with respect to homeomorphisms. Then $\mathbf{K}$-${\mathbf{\text{DCPO}}}_s$ is reflective in ${\mathbf{S}}_{\mathbf{K}}$-${\mathbf{\text{Poset}}}_s$. Therefore, if $\mathbf{K}\left(\mathrm{\Sigma }P\right)$ is a $\mathbf{K}$-dcpo for any poset $P$, then $\mathbf{K}$-${\mathbf{\text{DCPO}}}_s$ is reflective in ${\mathbf{\text{Poset}}}_s$.

By Theorems 5.6 and 9.5, we get the following three corollaries.

Corollary 9.8. ([39]) For a poset $P$, if ${\mathrm{Irr}}_c\left(\mathrm{\Sigma }P\right)$ is a $\mathbf{Sob}$-dcpo, then ${\mathbf{Sob}}_s\left(P\right)={\mathrm{Irr}}_c\left(\mathrm{\Sigma }P\right)$ with the canonical mapping ${\eta }_P:P\to {\mathbf{Sob}}_s\left(P\right)$, ${\eta }_P\left(x\right)={\mathrm{cl}}_c\sigma \left(P\right)\left\{x\right\}=\downarrow x$, is a ${\mathbf{Sob}}_s$-completion of $P$.

Corollary 9.9. ([39]) For a poset $P$, if $\mathrm{WD}\left(\mathrm{\Sigma }P\right)$ is a $\mathbf{WF}$-dcpo, then ${\mathbf{WF}}_s\left(P\right)=\mathrm{WD}\left(\mathrm{\Sigma }P\right)$ with the canonical mapping ${\eta }_P:P\to {\mathbf{WF}}_s\left(P\right)$, ${\eta }_P\left(x\right)={\mathrm{cl}}_{\sigma \left(P\right)}\left\{x\right\}=\downarrow x$, is a ${\mathbf{WF}}_s$-completion of $P$.

Corollary 9.10. Let $\mathbf{K}$ be a Keimel-Lawson category which is a full subcategory of ${\mathbf{Top}}_d$. For a poset $P$, if $\mathbf{K}\left(\mathrm{\Sigma }P\right)$ is a $\mathbf{K}$-dcpo, then ${\mathbf{K}}_s\left(P\right)=\mathbf{K}\left(\mathrm{\Sigma }P\right)$ with the canonical mapping ${\eta }_P:P\to {\mathbf{K}}_s\left(P\right)$, ${\eta }_P\left(x\right)={\mathrm{cl}}_{\sigma \left(P\right)}\left\{x\right\}=\downarrow x$, is a ${\mathbf{K}}_s$-completion of $P$.

Proposition 9.11. ([39]) Let $\mathbf{K}$ be a full subcategory of ${\mathbf{Top}}_d$ containing $\mathbf{Sob}$ which is adequate and closed with respect to homeomorphisms. For a non-$\mathbf{K}$ poset $P$, if ${\mathrm{Irr}}_c\left(\mathrm{\Sigma }P\right)=\left\{\overline{\left\{x\right\}}:x\in P\right\}\cup \left\{P\right\}$ and $P_{\mathrm{T}}$ is a $\mathbf{K}$-dcpo, then ${\mathbf{K}}_s\left(P\right)=P_{\mathrm{T}}$ with the canonical mapping ${\eta }_P:P\to P_{\mathrm{T}}$, ${\eta }_P\left(x\right)=x$, is a ${\mathbf{K}}_s$-completion of $P$.

By Corollary 6.9 and Theorem 9.5, we get the following result.

Corollary 9.12. ([39]) Let $P$ be a poset $P$. If ${\mathrm{Irr}}_c\left(\mathrm{\Sigma }P\right)=\left\{\overline{\left\{x\right\}}:x\in P\right\}\cup \left\{P\right\}$ and $\mathrm{\Sigma }{P}_{{\rm T}}$ is a sober space, then ${\mathbf{Sob}}_{\mathbf{s}}\left(P\right)=P_{{\rm T}}$ with the canonical mapping ${\eta }_P:P\to P_{{\rm T}}$, ${\eta }_P\left(x\right)=x$, is a ${\mathbf{Sob}}_{\mathbf{s}}$-completion of $P$.

Corollary 9.13. ([39]) Let $P$ be a poset $P$. If ${\mathrm{Irr}}_c\left(\mathrm{\Sigma }P\right)=\left\{\overline{\left\{x\right\}}:x\in P\right\}\cup \left\{P\right\}$ and $\mathrm{\Sigma }{P}_{{\rm T}}$ is a well-filtered space, then ${\mathbf{WF}}_s\left(P\right)=P_{\mathrm{T}}$ with the canonical mapping ${\eta }_P:P\to P_{\mathrm{T}}$, ${\eta }_P\left(x\right)=x$, is a ${\mathbf{WF}}_s$-completion of $P$.

10. K-reflections of Alexandroff spaces

In this section, we discuss the K-reflections of Alexandroff spaces and the K-completions of posets.

It is straightforward to verify the following lemma.

Lemma 10.1. ([39]) For a poset $P$, a $T_0$ space $Y$, and a mapping $f:\left(P,\alpha \left(P\right)\right)\to Y$, the following conditions are equivalent:

1. $f:\left(P,\alpha \left(P\right)\right)\to Y$ is continuous.

2. $f:P\to \mathrm{\Omega }Y$ is monotone.

3. $f:\left(P,\alpha \left(P\right)\right)\to \left(\mathrm{\Omega }Y,\alpha \left(\mathrm{\Omega }Y\right)\right)$ is continuous.

Lemma 10.2. ([39]) Let $\mathbf{K}$ be a full subcategory of ${\mathbf{Top}}_d$ containing $\mathbf{Sob}$ and $P$ a poset. Then $\mathbf{K}\left(\left(P,\alpha \left(P\right)\right)\right)=\mathrm{Id}P$.

Theorem 10.3. ([39]) Let $\mathbf{K}$ be a full subcategory of ${\mathbf{Top}}_d$ containing $\mathbf{Sob}$ which is adequate and closed with respect to homeomorphisms. Then for any poset $P$, the $\mathbf{K}$-reflection ${\left(\right)}^k$ of $\left(P,\alpha \left(P\right)\right)$ exists and it is a Scott space. More precisely, the Scott space $\mathrm{\Sigma }\mathrm{Id}P$ with the canonical mapping ${\varphi }_P:\left(P,\alpha \left(P\right)\right)\to \mathrm{\Sigma }\mathrm{Id}P$, $x\mapsto {\mathrm{cl}}_{\gamma \left(P\right)}\left\{x\right\}=\downarrow x$, is a $\mathbf{K}$-reflection of $\left(P,\alpha \left(P\right)\right)$.

Remark 10.4. ([39]) We can present a direct proof of Theorem 10.3.

Proof. Since $\mathrm{Id}P$ is an algebraic domain (cf. [4], Proposition I-4.10), by ([4], Proposition III-3.7) and Lemma 10.2, $\mathbf{K}\left(\left(P,\alpha \left(P\right)\right)\right)=\mathrm{Id}P$ and $\mathrm{\Sigma }\mathrm{Id}P$ is sober and hence a $\mathbf{K}$-space. Clearly, the canonical mapping ${\varphi }_P:\left(P,\alpha \left(P\right)\right)\to \mathrm{\Sigma }\mathbf{K}\left(\left(P,\alpha \left(P\right)\right)\right)=\mathrm{\Sigma }\mathrm{Id}P$, $x\mapsto {\mathrm{cl}}_{\alpha \left(P\right)}\left\{x\right\}=\downarrow x$, is continuous. Now we show that for each $\mathbf{K}$-space $Y$ and each continuous mapping $f:\left(P,\alpha \left(P\right)\right)\to Y$, there is a unique continuous mapping $f^{\ast }:\mathrm{\Sigma }\mathrm{Id}P\to Y$ such that $f^{\ast }\circ {\varphi }_P=f$, that is, the following diagram commutes.

We first prove the existence of $f^{\ast }$. Since $Y$ is a K-space, $Y$ is a $d$-space. Therefore, $\vee E$ exists in $Y$ for each directed subset $E$ of $Y$ (with the specialization order). As $f:\left(P,\alpha \left(P\right)\right)\to Y$ is continuous, $f:P\to \mathrm{\Omega }Y$ is monotone. Define a mapping $f^{\ast }:\mathrm{\Sigma }\mathrm{Id}P\to Y$ by $f^{\ast }\left(I\right)=\vee f\left(I\right)$ for each $I\in \mathrm{Id}P$. For every $\left\{I_d:d\in D\right\}\in \mathcal{D}\left(\mathrm{Id}P\right)$, we have that $f^{\ast }\left({\vee }_{\mathrm{Id}P}\left\{I_d:d\in D\right\}\right)=f^{\ast }\left({\cup }_{d\in D}{I}_d\right)=\vee f\left({\cup }_{d\in D}{I}_d\right)={\vee }_{d\in D}\vee f\left(I_d\right)={\vee }_{d\in D}{f}^{\ast }\left(I_d\right)$. By Lemmas 2.2 and 2.4, $f^{\ast }:\mathrm{\Sigma }\mathrm{Id}P\to Y$ is continuous. For each $x\in P$, since $f:P\to \mathrm{\Omega }Y$ is monotone, we have $f^{\ast }\left({\varphi }_P\left(x\right)\right)=\vee f\left(\downarrow x\right)=f\left(x\right)$, proving that $f^{\ast }\circ {\varphi }_P=f$.

Now we prove the uniqueness of $f^{\ast }$. Suppose that $g:\left(P,\alpha \left(P\right)\right)\to Y$ is another continuous mapping satisfying $g\circ \circ {\varphi }_P=f$. Then for each $I\in \mathrm{Id}P$, by Lemmas 2.2 and 2.4, we have that $g\left(I\right)=g\left({\cup }_{x\in I}\downarrow x\right)=g\left({\vee }_{\mathrm{Id}P}\left\{\downarrow x:x\in I\right\}\right)={\vee }_{x\in I}g\left(\downarrow x\right)={\vee }_{x\in I}f\left(x\right)=\vee f\left(I\right)=f^{\ast }\left(I\right)$, and hence $g=f^{\ast }$.

Therefore, $⟨\mathrm{\Sigma }\mathrm{Id}P,{\varphi }_P:\left(P,\alpha \left(P\right)\right)\to \mathrm{\Sigma }\mathrm{Id}P⟩$ is a K-reflection of $\left(P,\alpha \left(P\right)\right)$. □

From Theorem 10.3, we immediately deduce the following result.

Corollary 10.5. ([39]) Let $\mathbf{K}$ be a full subcategory of ${\mathbf{Top}}_d$ containing $\mathbf{Sob}$ which is adequate and closed with respect to homeomorphisms. Then for any poset $P$, the $\mathbf{K}$-completion of $P$ exists and it is the pair $⟨\mathrm{Id}P,{\varphi }_P⟩$, where ${\varphi }_P:P\to \mathrm{Id}P$ is defined by ${\varphi }_P\left(x\right)=\downarrow x$ for each $x\in P$.

Corollary 10.6. ([39]) Let $\mathbf{K}$ be a full subcategory of ${\mathbf{Top}}_d$ containing $\mathbf{Sob}$ which is adequate and closed with respect to homeomorphisms. Then $\mathbf{K}$-${\mathbf{\text{DCPO}}}_s$ is reflective in $\mathbf{\text{Poset}}$.

Finally, by Theorem 5.6 and Corollaries 10.5 and 10.6, we have the following three corollaries.

Corollary 10.7. ([39]) Let $P$ be a poset. Then the $d$-reflection of $\mathrm{\Gamma }P$, the well-filtered reflection of $\mathrm{\Gamma }P$ and the sobrification of $\mathrm{\Gamma }P$ agree. They all are the Scott space $\mathrm{\Sigma }\mathrm{Id}$ with the canonical mapping ${\varphi }_P:\mathrm{\Gamma }P\to P$.

Corollary 10.8. ([39]) Let $P$ be a poset. Then the $\mathbf{D}$-completion of $P$, the $\mathbf{WF}$-completion of $P$ and the $\mathbf{Sob}$-completion of $P$ agree. They all are the pair $⟨\mathrm{Id}P,{\varphi }_P⟩$, where ${\varphi }_P:P\to \mathrm{Id}P$ is defined by ${\varphi }_P\left(x\right)=\downarrow x$ for each $x\in P$.

Corollary 10.9. ([39]) ${\mathbf{\text{DCPO}}}_s$, $\mathbf{WF}$-${\mathbf{\text{DCPO}}}_s$ and $\mathbf{Sob}$-${\mathbf{\text{DCPO}}}_s$ all are reflective in $\mathbf{\text{Poset}}$.

11. Sobriety of Scott topology on dcpos

The following concept was first introduced in ([35], Definition 10.2.11) (see also [30]).

Definition 11.1. ([35]) A $T_0$ space $X$ is said to have property R if for any family $\left\{\uparrow F_i:i\in I\right\}\subseteq \mathbf{Fin}P$ and any $U\in \mathcal{O}\left(X\right)$, ${\cap }_{i\in I}\uparrow F_i\subseteq U$ implies ${\cap }_{i\in I_0}\uparrow F_i\subseteq U$ for some $I_0\in I^{\left(<\omega \right)}$. For a poset $P$, when $\mathrm{\Sigma }P$ has property R, we will simply say that $P$ has property R.

Proposition 11.2. ([35, 40]) For a poset $P$, consider the following conditions:

1. $\left(P,\lambda \left(P\right)\right)$ is compact.

2. $\left(P,\lambda \left(P\right)\right)$ is upper semicompact.

3. $P$ has property R.

4. For any filtered family $\left\{\uparrow F_i:i\in I\right\}\subseteq \mathbf{Fin}P$ and any $U\in \sigma \left(P\right)$, ${\cap }_{i\in I}\uparrow F_i\subseteq U$ implies $\uparrow F_j\subseteq U$ for some $j\in I$.

5. $P$ is a dcpo.

Then (1) $\Rightarrow$ (2) $\Rightarrow$ (3) $\Rightarrow$ (4) $\Rightarrow$ (5).

Theorem 11.3. ([35, 40]) For a poset $P$, the following conditions are equivalent:

1. $P$ has property R.

2. For any $\left\{x_i:i\in I\right\}\subseteq P$ and any $U\in \sigma \left(P\right)$ with ${\cap }_{i\in I}\uparrow x_i\subseteq U$, there is $I_0\in I^{\left(<\omega \right)}$ such that ${\cap }_{i\in I_0}\uparrow x_i\subseteq U$.

3. The compact saturated subsets of $\left(P,\omega \left(P\right)\right)$ are exactly the closed subsets of $\mathrm{\Sigma }P$.

4. Every Scott closet subset of $P$ is compact in $\mathrm{\Sigma }P$.

Proposition 11.4. ([35, 40]) Let $P$ be a poset. If $\mathrm{\Sigma }P$ is compact and ${\cap }_{x\in F}\uparrow x$ is compact in $\mathrm{\Sigma }P$ for each $F\in P^{\left(<\omega \right)}$, then the following two conditions are equivalent:

1. $\left(P,\lambda \left(P\right)\right)$ is compact.

2. $P$ has property R.

Clearly, if $\left(P,\lambda \left(P\right)\right)$ is compact, then $\left(P,\sigma \left(P\right)\right)$ is compact, and for each $F=\left\{x_1,x_2,\dots ,x_n\right\}\in P^{\left(<\omega \right)}$, $\uparrow x_1\cap \uparrow x_2\cap \dots \cap \uparrow x_n$ is a closed subset of $\left(P,\lambda \left(P\right)\right)$, whence it is compact in $\left(P,\lambda \left(P\right)\right)$, and consequently, compact in $\left(P,\sigma \left(P\right)\right)$. So Proposition 11.4 can be restated as the following one.

Proposition 11.5. ([35, 40]) For a poset, the following two conditions are equivalent:

1. $\left(P,\lambda \left(P\right)\right)$ is compact.

2. $\mathrm{\Sigma }P$ is compact, $\uparrow x_1\cap \uparrow x_2\cap \dots \cap \uparrow x_n$ is Scott compact (i.e., compact in $\mathrm{\Sigma }P$) for each $\left\{x_1,x_2,\dots ,x_n\right\}\in P^{\left(<\omega \right)}$, and $P$ has property R.

The following concept was introduced by Lawson, Wu, and Xi [21].

Definition 11.6. ([21]) A $T_0$ space $X$ is said to be ${\mathrm{\Omega }}^{\ast }$-compact if every closed subset of $X$ is compact in $\left(X,\omega \left(X\right)\right)$.

As $\left\{X\backslash \uparrow F:F\in \mathbf{Fin}X\right\}$ is a base of $\omega \left(X\right)$, we clearly have the following result.

Proposition 11.7. Let $X$ be a $T_0$ space. Then the following two conditions are equivalent:

1. $X$ has property R.

2. $X$ is ${\mathrm{\Omega }}^{\ast }$-compact.

Proposition 11.8. ([40]) Let $P$ be a dcpo for which $\mathrm{\Sigma }P$ is well-filtered and coherent. Then $P$ has property R.

In what follows, for a poset $P$, let ${\omega }^{\ast }\left(P\right)$ denote the family of all closed subsets of $\left(P,\omega \left(P\right)\right)$, namely, ${\omega }^{\ast }\left(P\right)=\left\{\varnothing ,P\right\}\cup \left\{\cap \mathcal{F}:\mathcal{F}\subseteq \mathbf{Fin}P\right\}$.

Lemma 11.9. ([40]) Let $P$ be a poset and $Q=\left({\omega }^{\ast }\left(P\right),\supseteq \right)$ (i.e., the order on $Q$ is the reverse inclusion order). Then $m:P\times P\to Q$, $\left(x,y\right)\mapsto \uparrow x\cap \uparrow y$, is Scott continuous.

Corollary 11.10. ([40]) Let $P$ be a poset and $Q=\left({\omega }^{\ast }\left(P\right),\supseteq \right)$. If $\mathrm{\Sigma }\left(P\times P\right)=\mathrm{\Sigma }P\times \mathrm{\Sigma }P$, then $m:\mathrm{\Sigma }P\times \mathrm{\Sigma }P\to \mathrm{\Sigma }Q$, $m\left(x,y\right)=\uparrow x\cap \uparrow y$, is continuous.

Corollary 11.11. ([40]) Let $P$ be a poset and $Q=\left({\omega }^{\ast }\left(P\right),\supseteq \right)$. If $\mathrm{\Sigma }P$ is core compact (especially, locally compact), then $m:\mathrm{\Sigma }P\times \mathrm{\Sigma }P\to \mathrm{\Sigma }Q$, $m\left(x,y\right)=\uparrow x\cap \uparrow y$, is continuous.

Using the Scott topology and the lower topology, we give the following useful characterization of property R.

Lemma 11.12. ([40]) Let $P$ be a poset and $Q=\left({\omega }^{\ast }\left(P\right),\supseteq \right)$. Then the following two conditions are equivalent:

1. $P$ has property R.

2. For any $U\in \sigma \left(P\right)$, $\mathrm{\Phi }\left(U\right)=\left\{C\in {\omega }^{\ast }\left(P\right):C\subseteq U\right\}\in \sigma \left(Q\right)$.

Theorem 11.13. ([40]) Let $P$ be a poset satisfying property R and $\mathrm{\Sigma }\left(P\times P\right)=\mathrm{\Sigma }P\times \mathrm{\Sigma }P$. Then $\mathrm{\Sigma }P$ is sober.

Corollary 11.14. ([40]) Let $P$ be a poset satisfying $\mathrm{\Sigma }\left(P\times P\right)=\mathrm{\Sigma }P\times \mathrm{\Sigma }P$. If $\mathrm{\Sigma }P$ is well-filtered and coherent. Then $\mathrm{\Sigma }P$ is sober.

Clasification

2000 MSC: 54D99; 54B30; 54B20; 06B35