Coincidence of the upper Vietoris topology and the Scott topology
aa r X i v : . [ m a t h . GN ] M a r Coincidence of the upper Vietoris topology and the Scott topology ✩ Xiaoquan Xu a, ∗ , Zhongqiang Yang b a School of Mathematics and Statistics, Minnan Normal University, Zhangzhou 363000, China b Department of Mathematics, Shantou University, Shantou 515063, China
Abstract
For a T space X , let K ( X ) be the poset of all compact saturated sets of X with the reverse inclusionorder. The space X is said to have property Q if for any K , K ∈ K ( X ), K ≪ K in K ( X ) iff K ⊆ int K .In this paper, we give several connections among the well-filteredness of X , the sobriety of X , the localcompactness of X , the core compactness of X , the property Q of X , the coincidence of the upper Vietoristopology and Scott topology on K ( X ), and the continuity of x
7→ ↑ x : X −→ Σ K ( X ) (where Σ K ( X ) isthe Scott space of K ( X )). It is shown that for a well-filtered space X for which its Smyth power space P S ( X ) is first-countable, the following three properties are equivalent: the local compactness of X , the corecompactness of X and the continuity of K ( X ). It is also proved that for a first-countable T space X in whichthe set of minimal elements of K is countable for any compact saturated subset K of X , the Smyth powerspace P S ( X ) is first-countable. For the Alexandroff double circle Y , which is Hausdorff and first-countable,we show that its Smyth power space P S ( Y ) is not first-countable. Keywords:
Well-filtered space; Local compactness; Smyth power space; Scott topology; First-countability
1. Introduction
In non-Hausdorff topology and domain theory, we encounter numerous links between topology and ordertheory. There are a lot of connections among well-filteredness, sobriety, local compactness and core com-pactness. The Hofmann-Mislove Theorem, the spectral theory of distributive continuous lattices and theduality theorem of continuous semilattices show some of the most important such connections (see [5, 7]).For a T space X , let K ( X ) be the poset of all compact saturated sets of X with the reverse inclusion order.The space X is said to have property Q if for any K , K ∈ K ( X ), K ≪ K in K ( X ) iff K ⊆ int K (cf.[5, Proposition I-1.24.2 and Proposition IV-2.19]). It is well-known that the local compactness of a T space X implies that its topology O ( X ) is a continuous lattice. The spectral theory of continuous lattices showsthat a sober space X for which O ( X ) is a continuous domain is locally compact, and if a well-filtered space X is locally compact, then K ( X ) is a continuous semilattice, but the converse fails in general. The dualitytheorem of continuous semilattices shows that for a sober space with property Q, K ( X ) is a continuoussemilattice iff X is locally compact (see [11, 10, 5, 7]). Thus the Lawson dual of K ( X ) may be properly”bigger” than O ( X ).The Smyth power spaces are very important structures in domain theory, which play a fundamental rolein modeling the semantics of non-deterministic programming languages. There naturally arises a question ofwhich topological properties are preserved by the Smyth power spaces. It was proved by Schalk [14] that theSmyth power space P S ( X ) of a sober space X is sober (see also [9, Theorem 3.13]), and the upper Vietoris ✩ This research was supported by the National Natural Science Foundation of China (Nos. 11661057, 11971287) and NSF ofJiangxi Province (20192ACBL20045) ∗ Corresponding author
Email addresses: [email protected] (Xiaoquan Xu), [email protected] (Zhongqiang Yang)
Preprint submitted to Topology and its applications March 17, 2020 opology (that is, the topology of Smyth power space) agrees with the Scott topology on K ( X ) if X is alocally compact sober space. Xi and Zhao [16] showed that a T space X is well-filtered iff P S ( X ) is a d -space.Recently, Brecht and Kawai [1] pointed out that P S ( X ) is second-countable for a second-countable T space X , and the first author and Zhao [20] proved that a T space X is well-filtered iff P S ( X ) is well-filtered.In this paper, we investigate some further connections among well-filteredness, sobriety, local compact-ness, core compactness, property Q, and coincidence of the upper Vietoris topology and the Scott topology.Especially, for a T space X , we discuss the following questions:(a) Under what conditions does the core compactness of a T space X imply the local compactness of X ?Does the continuity of K ( X ) of X equate to the local compactness of X ?(b) When does the upper Vietoris topology and the Scott topology on K ( X ) coincide?(c) Is the Smyth power space P S ( X ) of a first-countable T space again first-countable?For a T space X , we give several connections among the well-filteredness of X , the sobriety of X , thelocal compactness of X , the core compactness of X , the property Q of X , the coincidence of the upperVietoris topology and Scott topology on K ( X ), and the continuity of x
7→ ↑ x : X −→ Σ K ( X ) (where Σ K ( X )is the Scott space of K ( X )). It is shown that for a well-filtered space X for which P S ( X ) is first-countable,the following three properties are equivalent: the local compactness of X , the core compactness of X andthe continuity of K ( X ). Some known results are improved. It is proved that for a first-countable T space X in which the set of minimal elements of K is countable for any compact saturated subset K of X , theSmyth power space P S ( X ) is first-countable. For the Alexandroff double circle Y , which is Hausdorff andfirst-countable, we show that its Smyth power space P S ( Y ) is not first-countable.
2. Preliminary
In this section, we briefly recall some fundamental concepts and notations that will be used in the paper.Some basic properties of sober spaces, metric spaces and compact saturated sets are presented. For furtherdetails, we refer the reader to [5, 7, 14].For a poset P and A ⊆ P , let ↓ A = { x ∈ P : x ≤ a for some a ∈ A } and ↑ A = { x ∈ P : x ≥ a for some a ∈ A } . For x ∈ P , we write ↓ x for ↓{ x } and ↑ x for ↑{ x } . A subset A is called a lower set (resp.,an upper set ) if A = ↓ A (resp., A = ↑ A ). Define A ↑ = { x ∈ P : x is an upper bound of A in P } . Dually,define A ↓ = { x ∈ P : x is a lower bound of A in P } . The set A δ = ( A ↑ ) ↓ is called the cut generated by A .Let P ( <ω ) = { F ⊆ P : F is a nonempty finite set } , P ( ω ) = { F ⊆ P : F is a nonempty countable set } and Fin P = {↑ F : F ∈ P ( <ω ) } . For a nonempty subset A of P , define min( A ) = { a ∈ A : a is a minimalelement of A } . For a set X and A, B ⊆ X , A ⊂ B means that A ⊆ B but A = B , that is, A is a propersubset of B .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 D ( P ). I ⊆ P is called an ideal of P if I is a directed lowersubset of P . Let Id( P ) be the poset (with the order of set inclusion) of all ideals of P . Dually, we define theconcept of filters and denote the poset of all filters of P by Filt( P ). A filter of P is called principal if it hasa minimum element, that is, there is x ∈ P with F = ↑ x . P is called a directed complete poset , or dcpo forshort, provided that W D exists in P for any D ∈ D ( P ). P is called bounded complete if P is a dcpo and V A exists in P for any nonempty subset A of P .As in [5], the lower topology on a poset Q , generated by the complements of the principal filters of Q , isdenoted by ω ( Q ). A subset U of Q is Scott open if (i) U = ↑ U and (ii) for any directed subset D for which W D exists, W D ∈ U implies D T U = ∅ . All Scott open subsets of Q form a topology. This topology iscalled the Scott topology on Q and denoted by σ ( Q ). The space Σ Q = ( Q, σ ( Q )) is called the Scott space of Q . The topology generated by ω ( Q ) S σ ( Q ) is called the Lawson topology on Q and denoted by λ ( Q ).For a T space X , we use ≤ X to represent the specialization order of X , that is, x ≤ X y iff x ∈ { y } ).In the following, when a T space X is considered as a poset, the order always refers to the specializationorder if no other explanation. Let O ( X ) (resp., Γ( X )) be the set of all open subsets (resp., closed subsets)of X . A space X is locally hypercompact (see [4, 8]) if for each x ∈ X and each open neighborhood U of x ,2here is F ∈ X ( <ω ) such that x ∈ int ↑ F ⊆ ↑ F ⊆ U . Let | X | be the cardinality of X and ω = | N | , where N is the set of all natural numbersA T space X is called a d - space (or monotone convergence space ) if X (with the specialization order)is a dcpo and O ( X ) ⊆ σ ( X ) (cf. [5, 15]). Obviously, for a dcpo P , Σ P is a d -space. A nonempty subset A of a X is irreducible if for any { F , F } ⊆ Γ( X ), A ⊆ F S F implies A ⊆ F or A ⊆ F . Denote by Irr ( X )(resp., Irr c ( X )) the set of all irreducible (resp., irreducible closed) subsets of X . Clearly, every subset of X that is directed under ≤ X is irreducible. A space X is called sober , if for any F ∈ Irr c ( X ), there is a uniquepoint a ∈ X such that F = { a } . Clearly, sober spaces are T .For a dcpo P and A, B ⊆ P , we say A is way below B , written A ≪ B , if for each D ∈ D ( P ), W D ∈ ↑ B implies D T ↑ A = ∅ . For B = { x } , a singleton, A ≪ B is written A ≪ x for short. For x ∈ P , let w ( x ) = { F ∈ P ( <ω ) : F ≪ x } , ⇓ x = { u ∈ P : u ≪ x } and K ( P ) = { k ∈ P : k ≪ k } . Points in K ( P ) arecalled compact elements of P .For the following definition and related conceptions, please refer to [5, 7]. Definition 2.1.
Let P be a dcpo and X a T space.(1) P is called a continuous domain , if for each x ∈ P , ⇓ x is directed and x = W ⇓ x .(2) P is called an algebraic domain , if for each x ∈ P , K ( P ) T ↓ x is directed and x = W K ( P ) T ↓ x .(3) P is called a quasicontinuous domain , if for each x ∈ P , {↑ F : F ∈ w ( x ) } is filtered and ↑ x = T {↑ F : F ∈ w ( x ) } .(4) X is called core compact if O ( X ) is a continuous lattice . Lemma 2.2. ([5])
For a dcpo P , P is continuous iff for each x ∈ U ∈ σ ( P ) , there is u ∈ U such that x ∈ int σ ( P ) ↑ u ⊆ ↑ u ⊆ U . Lemma 2.3. ([6, 8])
Let P be a dcpo P . Then (1) P is quasicontinuous iff Σ P is locally hypercompact. (2) If P is a quasicontinuous domain, then Σ P is sober. A subset B of a T space X is called saturated if B equals the intersection of all open sets containingit (equivalently, B is an upper set in the specialization order). We shall use K ( X ) to denote the set of allnonempty compact saturated subsets of X and endow it with the Smyth preorder , that is, for K , K ∈ K ( X ), K ⊑ K iff K ⊆ K . Let S u ( X ) = {↑ x : x ∈ X } . Lemma 2.4. ([5])
Let X be a T space. For a nonempty family { K i : i ∈ I } ⊆ K ( X ) , W i ∈ I K i exists in K ( X ) iff T i ∈ I K i ∈ K ( X ) . In this case W i ∈ I K i = T i ∈ I K i . A topological space X is called well-filtered if X is T , and for any open set U and any K ∈ D ( K ( X )), T K⊆ U implies K ⊆ U for some K ∈K .We have the following implications (which can not be reversed):sobriety ⇒ well-filteredness ⇒ d -space.For a T space X , let OFilt( O (X)) = σ ( O ( X )) T Filt( O ( X )). U ⊆ O ( X ) is called an open filter if U ∈
OFilt( O (X)). For K ∈ K ( X ), let Φ( K ) = { U ∈ O ( X ) : K ⊆ U } . Then Φ( K ) ∈ OFilt( O (X)) and K = T Φ( K ). Obviously, Φ : K ( X ) −→ OFilt( O (X)) , K Φ( K ), is an order embedding.The single most important result about sober spaces is the Hofmann-Mislove Theorem (see [11] or [5,Theorem II-1.20 and Theorem II-1.21]). Theorem 2.5. (The Hofmann-Mislove Theorem)
For a T space X , the following conditions are equivalent: (1) X is a sober space. (2) For any
F ∈
OFilt( O ( X )) , there is K ∈ K ( X ) such that F = Φ( K ) . (3) For any
F ∈
OFilt( O ( X )) , F = Φ( T F ) .
3y the Hofmann-Mislove Theorem, a T space X is sober iff Φ : K ( X ) −→ OFilt( O (X)) is an orderisomorphism. Theorem 2.6. ([5, 7, 12])
For a T 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. For U ∈ O ( X ), let U = { K ∈ K ( X ) : K ⊆ U } . The upper Vietoris topology on K ( X ) is the topologygenerated by { U : U ∈ O ( X ) } as a base, and the resulting space is called the Smyth power space or upperspace of X and is denoted by P S ( X ) (cf. [8, 14]). Remark 2.7. ([8, 9, 14]) Let X be a T space. Then(1) The specialization order on P S ( X ) is the Smyth order (that is, ≤ P S ( X ) = ⊑ ).(2) The canonical mapping ξ X : X −→ P S ( X ), x
7→ ↑ x , is an order and topological embedding.(3) P S ( S u ( X )) is a subspace of P S ( X ) and X is homeomorphic to P S ( S u ( X )).For a nonempty subset C of a T X , it is easy to see that C is compact iff ↑ C ∈ K ( X ). Furthermore, wehave the following useful result (see, e.g., [3, pp.2068]). Lemma 2.8.
Let X be a T space and C ∈ K ( X ) . Then C = ↑ min( C ) and min( C ) is compact. For a metric space (
X, d ), x ∈ X and a positive number r , let B ( x, ε ) = { y ∈ Y : d ( x, y ) < r } bethe r - ball about x . For a set A ⊆ X and a positive number r , by the r - ball about A we mean the set B ( A, r ) = S a ∈ A B ( a, r ).The following two results are well-known (cf. [2]). Proposition 2.9.
Every metric space is perfectly normal and first-countable. Therefore, it is sober.
Proposition 2.10.
Let ( X, d ) be a metric space and K a compact set of X . Then for any open set U containing K , there is an r > such that K ⊆ B ( K, r ) ⊆ U .
3. Well-filtered spaces and locally compact spaces
Firstly, we give the following two known results (see, e.g., [5, 7, 16, 18]).
Lemma 3.1.
Let X be a well-filtered space. Then (1) For any
K ∈ D ( K ( X )) , T K ∈ K ( X ) and W K ( X ) K = T K . (2) P S ( X ) is a d -space, and hence the upper Vietoris topology is coarser than the Scott topology on K ( X ) . Lemma 3.2.
Let X be a T space. Then (1) K ( X ) is semilattice ( the semilattice operation being S ) . (2) Let K , K ∈ K ( X ) and consider the following assertions: (a) K ⊆ int K . (b) K ≪ K in K ( X ) .If X is well-filtered, then (a) ⇒ (b) , and if X is locally compact, then (b) ⇒ (a) . (3) If X is well-filtered and locally compact, then K ( X ) is a continuous semilattice. Definition 3.3. A T space X is said to have property Q if for any K , K ∈ K ( X ), K ≪ K in K ( X ) iff K ⊆ int K . 4t follows from Lemma 3.2 that every locally compact well-filtered space has property Q. Theorem 3.9below shows that for a well-filtered space X with the property Q, X is locally compact iff K ( X ) is acontinuous semilattice.The following result is a direct inference of the Hofmann-Mislove Theorem. Proposition 3.4. ([5])
Let X be a sober space. If K ( X ) is continuous, then the following two conditionsare equivalent: (1) X has property Q. (2) For any pair ( U , V ) ∈ σ ( O ( X )) × σ ( O ( X )) , U ≪ V in σ ( O ( X )) implies there is V ∈ V such that V ⊆ T U . Theorem 3.5. ([19])
Every first-countable well-filtered space is sober.
Theorem 3.6. ([13, 18, 19])
Every core compact well-filtered space is sober.
Corollary 3.7.
A well-filtered space is locally compact iff it is core compact.
By Theorem 3.6, Theorem 2.6 can be strengthened into the following one.
Theorem 3.8.
For a T 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. Now we give one of the main results of this paper.
Theorem 3.9.
For a well-filtered space X , the following conditions are equivalent: (1) X is locally compact. (2) K ( X ) is a continuous semilattice, and the upper Vietoris topology and the Scott topology on K ( X ) agree. (3) K ( X ) is a continuous semilattice, and ξ σX : X −→ Σ K ( X ) , x
7→ ↑ x , is continuous. (4) K ( X ) is a continuous semilattice, and X has property Q. (5) X is core compact. Proof . (1) ⇒ (2): By Theorem 2.6, Lemma 3.2 and [14, Lemma 7.26] (see Proposition 5.3 below).(2) ⇒ (3): By Remark 2.7.(3) ⇒ (1): For x ∈ U ∈ O ( X ), by Lemma 3.1, ↑ x ∈ U ∈ σ ( K ( X )), and hence by Lemma 2.2, there is K ∈ K ( X ) with ↑ x ∈ int σ ( K ( X )) ↑ K ( X ) K ⊆ ↑ K ( X ) K ⊆ U . Let V = ( ξ σX ) − (int σ ( K ( X )) ↑ K ( X ) K ). Then by thecontinuity of ξ σX , we have V ∈ O ( X ) and x ∈ V ⊆ K ⊆ U . Thus X is locally compact.(1) ⇒ (4): By Lemma 3.2.(4) ⇒ (1): Let x ∈ U ∈ O ( X ). Then by the continuity of K ( X ), ⇓ K ( X ) ↑ x is directed (note that theorder on K ( X ) is the reverse inclusion order) and ↑ x = W K ( X ) ⇓ K ( X ) ↑ x . It follows from Lemma 2.4 that ↑ x = W K ( X ) ⇓ K ( X ) ↑ x = T ⇓ K ( X ) ↑ x ⊆ U , and hence by the well-filteredness of X , there is K ∈⇓ K ( X ) ↑ x such that K ⊆ U . Since X has property Q, we have ↑ x ⊆ int K ⊆ K ⊆ U . Therefore, X is locally compact.(1) ⇔ (5): By Corollary 3.7 or Theorem 3.8.Theorem 3.9 can be restated as the following one. Theorem 3.10.
Let X be a well-filtered space such that K ( X ) is a continuous semilattice. Then the followingconditions are equivalent: (1) X is locally compact. (2) The upper Vietoris topology and the Scott topology on K ( X ) agree. ξ σX : X −→ Σ K ( X ) , x
7→ ↑ x , is continuous. (4) X has property Q. Corollary 3.11. ([5, Proposition IV-2.19])
Let X be a sober space having property Q. Then K ( X ) is acontinuous semilattice iff X is locally compact. The following example shows that for a well-filtered space X , when X lacks property Q, Theorem 3.9may not hold. It also shows that the well-filteredness of X and the continuity of K ( X ) together do not implythe sobriety of X in general. Example 3.12.
Let X be a uncountably infinite set and X coc the space equipped with the co-countabletopology (the empty set and the complements of countable subsets of X are open). Then(a) Γ( X coc ) = {∅ , X } S X ( ω ) and Irr ( X coc ) = Irr c ( X coc ) = { X } S {{ x } : x ∈ X } .(b) K ( X coc ) = X ( <ω ) \ {∅} and int K = ∅ for all K ∈ K ( X coc ).(c) K ( X coc ) is a dcpo and every element in K ( X coc ) is compact. Hence K ( X coc ) is an algebraic domain.(d) X coc is a well-filtered T space, but it not sober.(e) The upper Vietoris topology and the Scott topology on K (( X coc )) do not agree.(f) ξ σX coc : X coc −→ Σ K ( X coc ), x
7→ ↑ x , is not continuous.(g) X coc does not have property Q.(h) X coc is not locally compact and not first countable.The following example shows that even for a sober space X , when X lacks property Q, [5, PropositionIV-2.19]) (i.e., Corollary 3.11) may not hold. Example 3.13. ([5, Example II-1.25]) Let p be a point in β ( N ) \ N , where β ( N ) is the Stone-Cˇech com-pactification of the discrete space of natural numbers, and consider on X = N S { p } the induced topology.Then the space X is a non-discrete Hausdorff space, and hence a sober space. Every compact subset of X isfinite. Therefore, K ( X ) is an algebraic domain and X is not locally compact. By Theorem 3.9, X does nothave property Q and is not core compact. Furthermore, the upper Vietoris topology and the Scott topologyon K ( X ) do not agree, and the mapping ξ σX : X −→ Σ K ( X ), x
7→ ↑ x , is not continuous.By Example 3.12, Theorem 3.9 (or Theorem 3.10) strengthens [5, Proposition IV-2.19]) and shows thatthe converse of [5, Proposition IV-2.19]) holds in the following sense: for a well-filtered space X such thatK(X) is a continuous semilattice, X is locally compact iff X has property Q. Lemma 3.14.
For a dcpo P , ξ σ Σ P : Σ P −→ Σ K (Σ P ) is continuous. Proof . For any D ∈ D ( P ), we have ξ σ Σ P ( W D ) = ↑ W D = T d ∈ D ↑ d = W K (Σ P ) ξ σ Σ P ( D ) by Lemma 2.4. So ξ σ Σ P : Σ P −→ Σ K (Σ P ) is continuous.We get the following corollary from Theorem 3.9 and Lemma 3.14. Corollary 3.15.
For a dcpo P having the well-filtered Scott topology, the following conditions are equivalent: (1) Σ P is locally compact. (2) K (Σ P ) is a continuous semilattice, and the upper Vietoris topology and the Scott topology on K (Σ P ) agree. (3) K (Σ P ) is a continuous semilattice, and Σ P has property Q. (4) K (Σ P ) is a continuous semilattice. (5) Σ P is core compact. Definition 3.16.
Let P be a poset equipped with a topology τ .(1) ( P, τ ) is called upper semicompact , if ↑ x is compact for any x ∈ P , or equivalently, if ↑ x T A is compactfor any x ∈ P and A ∈ Γ((
P, τ )). 62) (
P, τ ) is called weakly upper semicompact if ↑ x T A is compact for any x ∈ P and A ∈ Irr c (( P, τ )).
Lemma 3.17. ([21])
For a dcpo P , if ( P, λ ( P )) is weakly upper semicompact ( especially, if ( P, λ ( P )) isupper semicompact or P is bounded complete ) , then ( P, σ ( P )) is well-filtered. By Corollary 3.11 and Lemma 3.17, we get the following corollary.
Corollary 3.18.
For a dcpo P , if ( P, λ ( P )) is weakly upper semicompact ( especially, if ( P, λ ( P )) is uppersemicompact or P is bounded complete ) , then the following conditions are equivalent: (1) Σ P is locally compact. (2) K (Σ P ) is a continuous semilattice, and the upper Vietoris topology and the Scott topology on K (Σ P ) agree. (3) K (Σ P ) is a continuous semilattice, and Σ P has property Q. (4) K (Σ P ) is a continuous semilattice. (5) Σ P is core compact.
4. First-countability of Smyth power spaces
Now we consider the following question: for a first-countable (resp., second-countable) space X , does itsSmyth power space P S ( X ) be first-countable (resp., second-countable)?First, we have the following result, which was indicated in the proof of [1, Proposition 6]. Theorem 4.1.
For a T space, the following two conditions are equivalent: (1) X is second-countable. (2) P S ( X ) is second-countable. Proof . (1) ⇒ (2): Let B ⊆ O ( X ) be a countable base of X and let B S = { n S i =1 U i : n ∈ N and U i ∈B for all 1 ≤ i ≤ n } . Then B S is countable. Now we show that B S is a base of P S ( X ). Let K ∈ K ( X ) and U ∈ O ( X ) with K ∈ U . Then for each k ∈ K , there is U k ∈ B with k ∈ U k ⊆ U . By the compactness of K , there is a finite subset { k , k , ..., k m } ⊆ K such that K ⊆ V = m S i =1 U k i ⊆ U , and hence V ∈ B S and K ∈ V ⊆ U . Thus B S is a base of P S ( X ), proving that P S ( X ) is second-countable.(2) ⇒ (1): As a subspace of P S ( X ), P S ( S u ( X )) is second-countable, and hence X is second-countablesince X is homeomorphic to P S ( S u ( X )).Next, we consider the first-countability. Since the first-countability is a hereditary property and any T space X is homeomorphic to P S ( S u ( X )), a subspace of P S ( S ( X )), we have the following result. Proposition 4.2.
Let X be a T space. If P S ( X ) is first-countable, then X is first-countable. Consider in the plane R two concentric circles C i = { ( x, y ) ∈ R : x + y = i } , where i = 1 ,
2, and theirunion X = C S C ; the projection of C onto C from the point (0 ,
0) is denoted by p . On the set X wegenerate a topology by defining a neighbourhood system { B ( z ) : z ∈ X } as follows: B ( z ) = { z } for z ∈ C and B ( z ) = { U j ( z ) : j ∈ N } for z ∈ C , where U j = V j S p ( V j \ { z } ) and V j is is the arc of C with centerat z and of length 1 /j . The space X is called the Alexandroff double circle (see [2, Example 3.1.26]).
Proposition 4.3. ([2])
Let X be the Alexandroff double circle. Then (1) X is Hausdorff and first-countable. (2) X is not separable, and hence not second-countable. (3) X is compact and locally compact. (4) C is a compact subspace of X . C is a discrete subspace of X . The following example shows that the converse of Proposition 4.2 fails in general.
Example 4.4.
Let X = C S C be the Alexandroff double circle. Then by Proposition 4.3, X is a compactHausdorff first-countable space and C ∈ K ( X ). Now we prove that P S ( X ) is not first-countable. First,for any open subset U ∈ O ( X ) with C ⊆ U , there is a family { U j = V n ( j ) S p ( V n ( j ) \ { z j } ) : j ∈ J } ofbasic open sets such that C ⊆ S j ∈ J U j ⊆ U , where V n ( j ) is the arc of C with center at z j and of length1 /n ( j ), and p is the projection of C onto C from the point (0 , C , there is afinite set { z j , z j , ..., z j n } ⊆ C such that C ⊆ n S i =1 U j i ⊆ U , and hence C \ U ⊆ { p ( z j ) , p ( z j ) , ..., p ( z j n ) } .Thus C \ U is finite. Suppose that { W n : n ∈ N } is a countable family of open sets containing C . Then C \ T n ∈ N W n = S n ∈ N ( C \ W n ) is countable. Choose x ∈ C T T n ∈ N W n and let V = X \ { x } . Then C ⊆ V ∈ O ( X ) but W n * V for all n ∈ N . Thus there is no countable base at C in P S ( X ), proving that P S ( X ) is not first-countable. Theorem 4.5.
Let X be a first-countable T space. If min( K ) is countable for any K ∈ K ( X ) , then P S ( X ) is first-countable. Proof . For each x ∈ X , by the first-countability of X , there exists a countable base B x at x . Let K ∈ K ( X ).Then by assumption min( K ) is countable. Let B K = { S c ∈ C ϕ ( c ) : C ∈ min( K ) ( <ω ) and ϕ ∈ Q c ∈ C B c } .Then B K is countable. Now we show that B K is a base at K . Suppose that U ∈ O ( X ) and K ∈ U . Thenmin( K ) ⊆ K ⊆ U . For each k ∈ min( K ), there is a ψ ( k ) ∈ B k with k ∈ ψ ( k ) ⊆ U . By the compactness ofmin( K ), there is a finite set { k , k , ..., k m } ⊆ min( K ) such that min( K ) ⊆ m S i =1 ψ ( k i ) ⊆ U . Let V = m S i =1 ψ ( k i ).Then K ⊆ V ⊆ U . It follows that V ∈ B K and K ∈ V ⊆ U , proving that B K is a base at K . Thus P S ( X ) is first-countable. Corollary 4.6.
Let X be a first-countable T space. If all compact subsets of X are countable, then P S ( X ) is first-countable. Proposition 4.7.
For a metric space ( X, d ) , P S (( X, d )) is first-countable. Proof . For K ∈ K (( X, d )), let B K = { B ( K, /n ) : n ∈ N } . Then by Proposition 2.10, B K = { B ( K, /n ) : n ∈ N } is a countable base at K in P S (( X, d )). Thus P S (( X, d )) is first-countable.
5. Coincidence of the upper Vietoris topology and Scott topology
For a well-filtered space X , from Theorem 3.9 we know that it is an important property that the upperVietoris topology agrees with the Scott topology on K ( X ). In this section we investigate the conditionsunder which the upper Vietoris topology coincides with the Scott topology on K ( X ). Proposition 5.1. ([8])
Let P be a dcpo. If Σ P is well-filtered and locally compact, then the upper Vietoristopology agrees the Scott topology on K (Σ P ) . From Lemma 2.3 and Proposition 5.1 we get the following result.
Proposition 5.2. ([9])
For a quasicontinuous domain P , the upper Vietoris topology agrees with the Scotttopology on K (Σ P ) . For a general T space X , Schalk [14] proved the following result. Proposition 5.3. ([14]) If X is a locally compact sober space, then the upper Vietoris topology and theScott topology on K ( X ) coincide.
8y Theorem 3.8 and Proposition 5.3, we have the following corollary.
Corollary 5.4. If X is a core compact well-filtered space, then the upper Vietoris topology and the Scotttopology on K ( X ) coincide. Theorem 5.5. ([1]) If X is a second-countable sober space, then the upper Vietoris topology and the Scotttopology on K ( X ) coincide. From Theorem 3.9, Theorem 3.5 and Theorem 5.5, we directly deduce the following two results.
Corollary 5.6. If X is a second-countable well-filtered space, then the upper Vietoris topology and the Scotttopology on K ( X ) coincide. Corollary 5.7.
For a second-countable well-filtered space X , the following conditions are equivalent: (1) X is locally compact. (2) K ( X ) is a continuous semilattice, and ξ σX : X −→ Σ K ( X ) , x
7→ ↑ x , is continuous. (3) K ( X ) is a continuous semilattice, and X has property Q. (4) K ( X ) is a continuous semilattice. (5) X is core compact. In [10] (see [5, Exercise V-5.25]), Hofmann and Lawson constructed a second-countable core compact T space X in which every compact subset has empty interior. So X is not locally compact and does not haveproperty Q. By Theorem 4.1, P S ( X ) is second-countable; and by Corollary 3.7 or Corollary 5.7, X is notwell-filtered.Now we give another main result of this paper. Theorem 5.8. If X is a well-filtered space and P S ( X ) is first-countable, then the upper Vietoris topologyagrees with and the Scott topology on K ( X ) . Proof . By Lemma 3.1, O ( P S ( X )) ⊆ σ ( K ( X )). Now we show that O ( P S ( X )) ⊇ σ ( K ( X )). Assume K ∈ U ∈ O ( P S ( X )). Since P S ( X ) is first-countable and { V : V ∈ O ( X ) } is a base of P S ( X ), we have a countablefamily { U n : n ∈ N } ⊆ O ( X ) such that { U n : n ∈ N } is a base at K in P S ( X ). We can assume that U ⊇ U ⊇ ... ⊇ U n ⊇ U n +1 ⊇ ... (otherwise, we replace U n with T ni =1 U i for each n ∈ N ). We claim that U n ⊆ U for some n ∈ N . Assume, on the contrary, that U m * U for all m ∈ N . For each m ∈ N , choose K m ∈ U m \ U , and let G m = K S S n ≥ m K n .Claim 1: G m ∈ K ( X ) for each m ∈ N .Suppose that { W j : j ∈ J } is an open cover of G m and let W J = S j ∈ J W j . Then W J ∈ O ( X )and K ⊆ W J . By the compactness of K , there is J ∈ J ( <ω ) such that K ⊆ W J = S j ∈ J W j . Since { U n : n ∈ N } is a base at K in P S ( X ), there is n o ∈ N such that K ∈ U n ⊆ W J , that is, K ⊆ U n ⊆ W J . As ( U n ) n ∈ N is a decreasing sequence, we have that K l ⊆ U l ⊆ U n ⊆ W J for all l ≥ n .By the compactness of n − S i = m K i , there is J ∈ J ( <ω ) such that n − S i = m K i ⊆ W J = S j ∈ J W j . Therefore, G m = ( K S S n ≥ n K n ) S n − S i = m K i ⊆ W J S W J . Thus G m ∈ K ( X ).Claim 2: G m ⊇ G m +1 for each m ∈ N .Claim 3: K = T m ∈ N G m .Clearly, K ⊆ T m ∈ N G m . Conversely, assume x K . Then K ∈ ( X \ ↓ x ), and whence there is m ∈ N such that K ∈ U m ⊆ ( X \ ↓ x ). It follows that x G m for all m ≥ m . Hence x T m ∈ N G m . Therefore, K = T m ∈ N G m .By the above three claims and Lemma 2.4, K = W K ( X ) { G m : m ∈ N } ∈ U ∈ σ ( K ( X )), and hence G q ∈ U for some q ∈ N . But then K n ∈ U for all n ≥ q , a contradiction.Therefore, K ∈ U n ⊆ U for some n ∈ N , and consequently, U ∈ O ( P S ( X )). It is thus proved that theupper Vietoris topology and the Scott topology on K ( X ) coincide.9y Theorem 4.5 and Theorem 5.8, we get the following result. Corollary 5.9.
Let X be a first-countable well-filtered space. If min( K ) is countable for any K ∈ K ( X ) ,then the upper Vietoris topology and the Scott topology on K ( X ) coincide. Corollary 5.10.
For a first-countable well-filtered space X in which all compact subsets are countable, theupper Vietoris topology and the Scott topology on K ( X ) agree. By Theorem 3.9, Theorem 5.8 and Corollary 5.9, we have the following three corollaries.
Corollary 5.11.
Let X be a well-filtered space for which P S ( X ) is first-countable. Then the followingconditions are equivalent: (1) X is locally compact. (2) K ( X ) is a continuous semilattice, and ξ σX : X −→ Σ K ( X ) , x
7→ ↑ x , is continuous. (3) K ( X ) is a continuous semilattice, and X has property Q. (4) K ( X ) is a continuous semilattice. (5) X is core compact. Corollary 5.12.
Let X be a first-countable well-filtered space for which min( K ) is countable for any K ∈ K ( X ) . Then the following conditions are equivalent: (1) X is locally compact. (2) K ( X ) is a continuous semilattice, and ξ σX : X −→ Σ K ( X ) , x
7→ ↑ x , is continuous. (3) K ( X ) is a continuous semilattice, and X has property Q. (4) K ( X ) is a continuous semilattice. (5) X is core compact. Corollary 5.13.
Let X be a first-countable well-filtered space for which all compact subsets of X are count-able. Then the following conditions are equivalent: (1) X is locally compact. (2) K ( X ) is a continuous semilattice, and ξ σX : X −→ Σ K ( X ) , x
7→ ↑ x , is continuous. (3) K ( X ) is a continuous semilattice, and X has property Q. (4) K ( X ) is a continuous semilattice. (5) X is core compact. By Proposition 2.9, Lemma 4.7 and Theorem 5.8, we get the following two results.
Corollary 5.14.
For a metric space ( X, d ) , the upper Vietoris topology coincides with the Scott topology on K (( X, d )) . Corollary 5.15.
For a metric space ( X, d ) , the following conditions are equivalent: (1) ( X, d ) is locally compact. (2) K ( X ) is a continuous semilattice, and ξ σ ( X,d ) : ( X, d ) −→ Σ K (( X, d )) , x
7→ { x } , is continuous. (3) K (( X, d )) is a continuous semilattice, and ( X, d ) has property Q. (4) K (( X, d )) is a continuous semilattice. (5) ( X, d ) is core compact. Let R be the set of all real numbers. R endowed with the topology taking the family { [ x, y ) : x < y } asa base is called the Sorgenfrey line and denoted by R l . Dually, we endow R with the topology generatedby { ( x, y ] : x < y } as a base, and denote the resulting space by R r . A subset A ⊆ R l is called bounded if A ⊆ [ − n, n ] for some n ∈ N . As one of the ”universal counterexamples” in general topology, R l poses manyimportant topological properties (cf. [2, 7]). In particular, the Sorgenfrey line has the following properties(cf. [2]). 10 roposition 5.16. (1) R l is perfectly normal, first-countable and separable. (2) R l is not second-countable. (3) R l is neither compact nor locally compact. (4) Every compact subset of R l is countable. Lemma 5.17. ([17])
For a subset A of R l , the following two conditions are equivalent: (1) A is compact in R l . (2) A is a bounded closed subset of R l , and A has no accumulation point in R r ( that is, there is no point x ∈ R such that x ∈ cl R r ( A \ { x } )) . Example 5.18.
Consider the Sorgenfrey line R l . Then by Theorem 3.9, Corollary 4.6, Theorem 5.8,Proposition 5.16 and Lemma 5.17, we have(1) R l is first-countable and Hausdorff, and hence sober.(2) P S ( R l ) is first-countable.(3) int K = ∅ for any K ∈ K ( R l ), and whence P S ( R l ) is not locally compact.(4) the upper Vietoris topology and the Scott topology on K ( R l ) agree.(5) K K for any K , K ∈ K ( R l ), so K ( R l ) is not continuous and R l has property Q.(6) ξ σ R l : R l −→ Σ K ( R l ), x
7→ { x } , is continuous.(7) R l is not core compact. Example 5.19.
Let X be the space in Example 3.13. Then we have (a) | X | = ω and X is sober; (b) K ( X ) = X ( <ω ) \ {∅} ; (c) K ( X ) is an algebraic semilattice; and (d) the upper Vietoris topology and theScott topology on K ( X ) does not coincide or, equivalently, σ ( K ( X )) * O ( P S ( X )). By Proposition 4.2 (orTheorem 5.8) and Corollary 5.10, Neither P S ( X ) nor X is first-countable (cf. [2, Corollary 3.6.17]).Finally, we pose the following question. Question 5.20.
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