ω -Rudin spaces, well-filtered determined spaces and first-countable spaces
aa r X i v : . [ m a t h . GN ] A ug ω -Rudin spaces, well-filtered determined spaces and first-countable spaces ✩ Xiaoquan Xu a, ∗ , Chong Shen b , Xiaoyong Xi c , Dongsheng Zhao d a School of Mathematics and Statistics, Minnan Normal University, Zhangzhou 363000, China b School of Mathematical Sciences, Nanjing Normal University, Nanjing 210046, China c School of mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, China d Mathematics and Mathematics Education, National Institute of Education Singapore,Nanyang Technological University, 1 Nanyang Walk, Singapore 637616
Abstract
We investigate some versions of d -space, well-filtered space and Rudin space concerning various countabilityproperties. The main results include: (i) if the sobrification of a T space X is first-countable, then X isan ω -Rudin space; (ii) every ω -well-filtered space is sober if its sobrification is first-countable; (iii) if a T space is second-countable or first-countable and with a countable underlying set, then it is a ω -Rudin space;(iv) every first-countable T space is well-filtered determined; (v) every irreducible closed subset in a first-countable ω -well-filtered space is countably-directed; (vi) every first-countable ω -well-filtered ω ∗ - d -space issober. Keywords:
First-countability; Sober space; ω -Rudin space; ω -Well-filtered space; ω ∗ - d -Space;Countably-directed set
1. Introduction
In [17, 20, 21], we introduced and studied the Rudin spaces, well-filtered determined spaces and ω -well-filtered spaces. Some relationships and links among these new non-Hausdorff topological properties and thewell studied sobriety and well-filteredness were uncovered. In [21], it was proved that in a first-countable ω -well-filtered space X , every irreducible closed subset of X is directed under the specialization order of X .It follows immediately that every first-countable ω -well-filtered d -space is sober.In the current paper, we continue studying some aspects of d -space, well-filtered space and Rudin spacesconcerning countability. Employing countably-directed sets, we define the ω ∗ - d -spaces and ω ∗ -well-filteredspaces. It is proved that if the sobrification of a T space X is first-countable, then X is an ω -Rudin space.Therefore, every ω -well-filtered space is sober if it has a first countable sobrification. From these, we obtainthat if a T space X is second-countable or first-countable with a countable underlying set, then X is an ω -Rudin space, and X is sober if it is additionally ω -well-filtered. Another major result obtained is thatevery first-countable T space is well-filtered determined. In each first-countable ω -well-filtered space, everyirreducible closed subset is proved to be countably-directed, hence every first-countable ω -well-filtered ω ∗ - d -space is sober. We also prove that a T space Y is ω ∗ -well-filtered iff its Smyth power space is ω ∗ -well-filterediff its Smyth power space is an ω ∗ - d -space. The work presented here enriched the theory of non-Hausdorfftopological spaces and lead to some nontrivial open problems for further investigation. ✩ This research was supported by the National Natural Science Foundation of China (Nos. 11661057, 11361028, 61300153,11671008, 11701500, 11626207); the Natural Science Foundation of Jiangxi Province, China (No. 20192ACBL20045); NSFProject of Jiangsu Province, China (BK20170483); and NIE ACRF (RI 3/16 ZDS), Singapore ∗ Corresponding author
Email addresses: [email protected] (Xiaoquan Xu), [email protected] (Chong Shen), [email protected] (Xiaoyong Xi), [email protected] (Dongsheng Zhao)
Preprint submitted to Elsevier August 26, 2020 . Preliminary
In this section, we briefly recall some fundamental concepts and notations that will be used in the paper.Some basic properties of irreducible sets and compact saturated sets are presented.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 ). Let P ( <ω ) = { F ⊆ P : F is a nonempty finite set } . For a set X , | X | will denote the cardinality of X . Let N denotes the set of all natural numbers with the usual order and ω = | N | .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 ). A subset I ⊆ P is called an ideal of P if I is a directedlower subset of P . Let Id( P ) be the poset (with the order of set inclusion) of all ideals of P . Dually, wedefine the notion of filters and denote the poset of all filters of P by Filt( P ). A poset P is called a directedcomplete poset , or dcpo for short, if for any D ∈ D ( P ), W D exists in P .As in [3], the upper topology on a poset Q , generated by the complements of the principal ideals of Q , isdenoted by υ ( Q ). A subset U of Q is Scott open if (i) U = ↑ U and (ii) for any directed subset D ⊆ Q with W D existing, W D ∈ U implies D ∩ U = ∅ . All Scott open subsets of Q form a topology, called the Scotttopology on Q and denoted by σ ( Q ). The space Σ Q = ( Q, σ ( Q )) is called the Scott space of Q . The uppersets of Q form the ( upper ) Alexandroff topology α ( Q ).For a T space X , we use ≤ X to denote the specialization order on X : x ≤ X y iff x ∈ { y } ). In thefollowing, when a T space X is considered as a poset, the partial order always means the specializationorder provided otherwise indicated. Let O ( X ) (resp., C ( X )) be the set of all open subsets (resp., closedsubsets) of X , and let S u ( X ) = {↑ x : x ∈ X } . Let S c ( X ) = {{ x } : x ∈ X } and D c ( X ) = { D : D ∈ D ( X ) } .A nonempty subset A of X is irreducible if for any { F , F } ⊆ C ( X ), A ⊆ F ∪ 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 .Every directed subset of X is irreducible. X is called sober , if for any F ∈ Irr c ( X ), there is a unique point a ∈ X such that F = { a } . Remark 2.1.
In a T space X , if x ∈ X and A ⊆ X such that A = { x } , then W A exists in X and x = W A .The following two lemmas on irreducible sets are well-known. Lemma 2.2.
Let X be a space and Y a subspace of X . Then the following conditions are equivalent for asubset A ⊆ Y : (1) A is an irreducible subset of Y . (2) A is an irreducible subset of X . (3) cl X A is an irreducible subset of X . Lemma 2.3. If f : X −→ Y is continuous and A ∈ Irr ( X ) , then f ( A ) ∈ Irr ( Y ) . A T space X is called a d-space (or monotone convergence space ) if X (with the specialization order) isa dcpo and O ( X ) ⊆ σ ( X ) (cf. [3, 18]). Definition 2.4. ([21]) A T space X is called a directed closure space , DC space for short, if Irr c ( X ) = D c ( X ),that is, for each A ∈ Irr c ( X ), there exists a directed subset of X such that A = D .For any topological space X , G ⊆ X and A ⊆ X , let G A = { G ∈ G : G T A = ∅} and G A = { G ∈G : G ⊆ A } . The symbols G A and G A will be simply written as A and A respectively if no ambiguityoccurs. The lower Vietoris topology on G is the topology that has { U : U ∈ O ( X ) } as a subbase, and theresulting space is denoted by P H ( G ). If G ⊆
Irr ( X ), then { G U : U ∈ O ( X ) } is a topology on G . The space P H ( C ( X ) \ {∅} ) is called the Hoare power space or lower space of X and is denoted by P H ( X ) for short (cf.[16]). Clearly, P H ( X ) = ( C ( X ) \ {∅} , υ ( C ( X ) \ {∅} )). So P H ( X ) is always sober (see [24, Corollary 4.10] or[21, Proposition 2.9]). The upper Vietoris topology on G is the topology that has { G U : U ∈ O ( X ) } as abase, and the resulting space is denoted by P S ( G ). 2 emark 2.5. Let X be a T space.(1) If S c ( X ) ⊆ G , then the specialization order on P H ( G ) is the set inclusion order, and the canonicalmapping η X : X −→ P H ( G ), given by η X ( x ) = { x } , is an order and topological embedding (cf. [3, 6, 16]).(2) The space X s = P H ( Irr c ( X )) with the canonical mapping η X : X −→ X s is the sobrification of X (cf.[3, 6]).A subset A of a space X is called saturated if A equals the intersection of all open sets containingit (equivalently, A 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 . The space P S ( K ( X )), denoted shortly by P S ( X ), is called the Smyth power space or upper space of X (cf. [7, 16]). It is easy to verify that the specialization order on P S ( X ) is the Smythorder (that is, ≤ P S ( X ) = ⊑ ). The canonical mapping ξ X : X −→ P S ( X ), x
7→ ↑ x , is an order and topologicalembedding (cf. [7, 8, 16]). Clearly, P S ( S u ( X )) is a subspace of P S ( X ) and X is homeomorphic to P S ( S u ( X )). Lemma 2.6. ([3])
Let X be a T space. For any 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 . Lemma 2.7. ([10, 16])
Let X be a T space. (1) If K ∈ K ( P S ( X )) , then S K ∈ K ( X ) . (2) The mapping S : P S ( P S ( X )) −→ P S ( X ) , K 7→ S K , is continuous. A T space X is called well-filtered if it is T , and for any open set U and filtered family K ⊆ K ( X ), T K⊆ U implies K ⊆ U for some K ∈K . Remark 2.8.
The following implications are well-known (cf. [3]):sobriety ⇒ well-filteredness ⇒ d -space.
3. Topological Rudin’s Lemma, Rudin spaces and well-filtered determined spaces
Rudin’s Lemma is a useful tool in topology and plays a crucial role in domain theory (see [1, 3-9, 20-21,23]). Heckmann and Keimel [8] presented the following topological variant of Rudin’s Lemma.
Lemma 3.1. (Topological Rudin’s Lemma)
Let X be a topological space and A an irreducible subset of theSmyth power space P S ( X ) . Then every closed set C ⊆ X that meets all members of A contains a minimalirreducible closed subset A that still meets all members of A . Applying Lemma 3.1 to the Alexandroff topology on a poset P , one obtains the original Rudin’s Lemma(see [15]). Corollary 3.2. (Rudin’s Lemma)
Let P be a poset, C a nonempty lower subset of P and F ∈
Fin P afiltered family with F ⊆ C . Then there exists a directed subset D of C such that F ⊆ ↓ D . For a T space X and K ⊆ K ( X ), let M ( K ) = { A ∈ C ( X ) : K T A = ∅ for all K ∈ K} (that is, K ⊆ A )and m ( K ) = { A ∈ C ( X ) : A is a minimal member of M ( K ) } .By the proof of [8, Lemma 3.1], we have the following result. Lemma 3.3.
Let X be a T space and K ⊆ K ( X ) . If C ∈ M ( K ) , then there is a closed subset A of C suchthat C ∈ m ( K ) . In [17, 21], based on topological Rudin’s Lemma, Rudin spaces and well-filtered determined spaces wereintroduced and studied. 3 efinition 3.4. ([17, 21]) Let X be a T space. A nonempty subset A of X is said to have the Rudinproperty , if there exists a filtered family
K ⊆ K ( X ) such that A ∈ m ( K ) (that is, A is a minimal closed setthat intersects all members of K ). Let RD ( X ) = { A ∈ C ( X ) : A has Rudin property } . The sets in RD ( X )will also be called Rudin sets . The space X is called a Rudin space , RD space for short, if Irr c ( X ) = RD ( X ),that is, every irreducible closed set of X is a Rudin set. The category of all Rudin spaces with continuousmappings is denoted by Top r . Definition 3.5. ([21]) A subset A of a T space X is called a well-filtered determined set , WD set for short,if for any continuous mapping f : X −→ Y to a well-filtered space Y , there exists a unique y A ∈ Y suchthat f ( A ) = { y A } . Denote by WD ( X ) the set of all closed well-filtered determined subsets of X . X is calleda well-filtered determined space, WD space for short, if all irreducible closed subsets of X are well-filtereddetermined, that is, Irr c ( X ) = WD ( X ). Proposition 3.6. ([21]) Let X be a T space. Then D c ( X ) ⊆ RD ( X ) ⊆ WD ( X ) ⊆ Irr c ( X ) . Corollary 3.7. ([21]) Sober ⇒ DC ⇒ RD ⇒ WD . A topological space X is locally hypercompact if for each x ∈ X and each open neighborhood U of x ,there is ↑ F ∈ Fin X such that x ∈ int ↑ F ⊆ ↑ F ⊆ U (cf. [2]). A space X is called core - compact if ( O ( X ) , ⊆ )is a continuous lattice (cf. [3]). Theorem 3.8. ([2]) Let X be a locally hypercompact T space and A ∈ Irr ( X ) . Then there exists a directedsubset D ⊆ ↓ A such that A = D . Therefore, X is a DC space, and hence a Rudin space and a WD space. Theorem 3.9. ([21]) Every locally compact T space is a Rudin space. Theorem 3.10. ([21]) Every core-compact T space is well-filtered determined. By Theorem 3.10, we immediately deduce the following.
Corollary 3.11. ([13, 21]) Every core-compact well-filtered space is sober.
At the moment, it is still not sure wether every core-compact T space is a Rudin space. ω - d -Spaces and ω -well-filtered spaces For a T space X , let D ω ( X ) = { D ⊆ X : D is countable and directed } and D ωc ( X ) = { D : D ∈ D ω ( X ) } . Definition 4.1. ([20]) A poset P is called an ω -dcpo , if for any D ∈ D ω ( P ), W D exists. Lemma 4.2. ([20]) Let P be a poset and D ∈ D ω ( P ) . Then there exists a countable chain C ⊆ D suchthat D = ↓ C . Hence, W C exists and W C = W D whenever W D exists. By Lemma 4.2, a poset P is an ω -dcpo iff for any countable chain C of P , W C exists. Definition 4.3. ([20]) Let P be a poset. A subset U of P is called ω -Scott open if (i) U = ↑ U , and (ii)for any countable directed set D , W D ∈ U implies that D ∩ U = ∅ . All ω -Scott open sets form a topologyon P , denoted by σ ω ( P ) and called the ω -Scott topology . The space Σ ω P = ( P, σ ω ( P )) is called the ω -Scottspace of P .Clearly, σ ( P ) ⊆ σ ω ( P ). The converse need not be true, see Example 7.5 in Section 7. Definition 4.4. ([20]) A T space X is called an ω - d -space (or an ω -monotone convergence space ) if forany D ∈ D ω ( X ), the closure of D has a generic point, equivalently, if D ωc ( X ) = S c ( X ).Some characterizations of ω - d -spaces were given in [20, Proposition 3.7]4 efinition 4.5. ([20]) A T space X is called ω -well-filtered , if for any countable filtered family { K i : i <ω } ⊆ K ( X ) and U ∈ O ( X ), it holds that \ i<ω K i ⊆ U ⇒ ∃ i < ω, K i ⊆ U. By Lemma 4.2, we have the following result.
Proposition 4.6. ([20]) A T space X is ω -well-filtered iff for any countable descending chain K ⊇ K ⊇ K ⊇ . . . ⊇ K n ⊇ . . . of compact saturated subsets of X and U ∈ O ( X ) , the following implication holds: \ i<ω K i ⊆ U ⇒ ∃ i < ω, K i ⊆ U. It is easy to check that every ω -well-filtered space is an ω - d -space (see [20, Proposition 3.11]). In [20],we introduced and studied two new classes of closed subsets in T spaces - ω -Rudin sets and ω -well-filtereddetermined closed sets. The ω -Rudin sets lie between the class of all closures of countable directed subsetsand that of ω -well-filtered determined closed sets, and ω -well-filtered determined closed sets lie between theclass of all ω -Rudin sets subsets and that of irreducible closed subsets. Definition 4.7. ([20]) Let X be a T space. A nonempty subset A of X is said to have the ω - Rudin property ,if there exists a countable filtered family
K ⊆ K ( X ) such that A ∈ m ( K ) (that is, A is a minimal closedset that intersects all members of K ). Let RD ω ( X ) = { A ∈ C ( X ) : A has ω -Rudin property } . The sets in RD ω ( X ) will also be called ω - Rudin sets . The space X is called ω - Rudin space , if
Irr c ( X ) = RD ω ( X ), thatis, all irreducible closed subsets of X are ω -Rudin sets. Definition 4.8. ([20]) A subset A of a T space X is called an ω - well - filtered determined set , WD ω set forshort, if for any continuous mapping f : X −→ Y to an ω -well-filtered space Y , there exists a unique y A ∈ Y such that f ( A ) = { y A } . Denote by WD ω ( X ) the set of all closed ω -well-filtered determined subsets of X .The space X is called ω - well - filtered determined , ω - WD space for short, if Irr c ( X ) = WD ω ( X ), that is, allirreducible closed subsets of X are ω -well-filtered determined. Proposition 4.9. ([20]) Let X be a T space. Then S c ( X ) ⊆ D ωc ( X ) ⊆ RD ω ( X ) ⊆ WD ω ( X ) ⊆ Irr c ( X ) . By Proposition 4.9, every ω -Rudin space is ω -well-filtered determined. Definition 4.10. A T space X is called a countable directed closure space , ω - DC space for short, if Irr c ( X ) = D ωc ( X ), that is, for each A ∈ Irr c ( X ), there exists a countable directed subset of X such that A = D . Theorem 4.11. ([20]) For a T space X , the following conditions are equivalent: (1) X is sober. (2) X is an ω - DC and ω - d -space. (3) X is an ω - DC and ω -well-filtered space. (4) X is an ω -Rudin and ω -well-filtered space. (5) X is an ω -well-filtered determined and ω -well-filtered space. ω ∗ -Scott topologies and ω ∗ - d -spaces We now introduce and study two new types of spaces.
Definition 5.1.
A nonempty subset D of a poset P is called countably-directed if every nonempty countablesubset of D have an upper bound in D . The set of all countably-directed sets of P is denoted by D ω ∗ ( P ).The poset P is called a countably-directed complete poset , or ω ∗ - dcpo for short, if for any D ∈ D ω ∗ ( P ), W D exists in P . 5learly, {↓ x : x ∈ P } ⊆ D ω ∗ ( P ) ⊆ D ( P ). Example 5.2.
For the countable chain N (with the usual order of natural numbers), ( N ( <ω ) , ⊆ ) is directedin 2 N , but not countably-directed. Definition 5.3.
A subset U of a poset P is ω ∗ - Scott open if (i) U = ↑ U and (ii) for any countably-directedsubset D with W D existing, W D ∈ U implies D ∩ U = ∅ . All ω ∗ -Scott open subsets of P form a topology,called the ω ∗ - Scott topology on Q and denoted by σ ω ∗ ( Q ). Let Σ ω ∗ Q = ( Q, σ ω ∗ ( Q )).Clearly, υ ( P ) ⊆ σ ( P ) ⊆ σ ω ∗ ( P ) ⊆ α ( P ). In general, σ ( P ) = σ ω ∗ ( P ) as shown in Example 7.5 below.It is straightforward to verify the following two results (cf. [3, Proposition II-2.1]). Proposition 5.4.
A continuous function f : X → Y from an ω ∗ - d -space X to any T space Y preservescountably-directed sups in the specialization orders. Proposition 5.5.
Let
P, Q be posets and f : P → Q . Then the following two conditions are equivalent: (1) f : Σ ω ∗ P → Σ ω ∗ Q is continuous. (2) f preserves countably-directed sups, that is, for every D ∈ D ω ∗ ( P ) for which W D exists, W f ( D ) existsand f ( ∨ D ) = ∨ f ( D ) . Definition 5.6. A T space X is said to be an ω ∗ - d -space (or an ω ∗ -monotone convergence space ), if everysubset D countably-directed relative to the specialization order of X has a sup, and the relation sup D ∈ U for any open set U of X implies D ∩ U = ∅ .For a T space X (endowed with the specialization order), let D ω ∗ c ( X ) = { D : D ∈ D ω ∗ ( X ) } . Then S c ( X ) ⊆ D ω ∗ c ( X ) ⊆ D c ( X ). Proposition 5.7.
For a T space X , the following conditions are equivalent: (1) X is an ω ∗ - d -space. (2) D ω ∗ c ( X ) = S c ( X ) , that is, for any D ∈ D ω ∗ ( X ) , the closure of D has a (unique) generic point. (3) X (with the specialization order ≤ X ) is an ω ∗ -dcpo and O ( X ) ⊆ σ ω ∗ ( X ) . (4) For any D ∈ D ω ∗ ( X ) and U ∈ O ( X ) , T d ∈ D ↑ d ⊆ U implies ↑ d ⊆ U ( i.e., d ∈ U ) for some d ∈ D . (5) For any D ∈ D ω ∗ ( X ) and A ∈ C ( X ) , if D ⊆ A , then A ∩ T d ∈ D ↑ d = ∅ . (6) For any D ∈ D ω ∗ ( X ) and A ∈ Irr c ( X ) , if D ⊆ A , then A ∩ T d ∈ D ↑ d = ∅ . (7) For any D ∈ D ω ∗ ( X ) , D ∩ T d ∈ D ↑ d = ∅ . Proof . (1) ⇒ (2): Let D ∈ D ω ∗ ( X ). Then by (1), W D exists and the relation W D ∈ U for any open set U of X implies D ∩ U = ∅ . Therefor, D = { sup D } .(2) ⇒ (3): For each D ∈ D ω ∗ ( X ) and A ∈ C ( X ) with D ⊆ A , by condition (2) there is x ∈ X suchthat D = { x } , and consequently, W D = x and W D ∈ A since D ⊆ A . Thus X is an ω ∗ -dcpo and O ( X ) ⊆ σ ω ∗ ( X ).(3) ⇒ (4): Suppose that D ∈ D ω ∗ ( X ) and U ∈ O ( X ) with T d ∈ D ↑ d ⊆ U . Then by condition (3), ↑ W D = T d ∈ D ↑ d ⊆ U ∈ σ ω ∗ ( X ). Therefore, W D ∈ U , and whence d ∈ U for some d ∈ D .(4) ⇒ (5): If A ∩ T d ∈ D ↑ d = ∅ , then T d ∈ D ↑ d ⊆ X \ A . By condition (4), ↑ d ⊆ X \ A for some d ∈ D , whichis in contradiction with D ⊆ A .(5) ⇒ (6) ⇒ (7): Trivial.(7) ⇒ (1): For each D ∈ D ω ∗ ( X ) and A ∈ C ( X ) with D ⊆ A , by condition (6), we have D ∩ T d ∈ D ↑ d = ∅ .Select an x ∈ D ∩ T d ∈ D ↑ d . Then D ⊆ ↓ x ⊆ D , and hence D = ↓ x . Thus X is an ω ∗ - d -space.6y Proposition 5.7, every d -space is an ω - d -space, and for any ω ∗ -dcpo P , Σ ω ∗ P is an ω ∗ - d -space. Let Q = ( N ( <ω ) , ⊆ ). Then Q is an ω ∗ -dcpo but not a dcpo. If P is any ω ∗ -dcpo but not a dcpo, then Σ ω ∗ ( P )is an ω ∗ - d -space but not a d -space. Definition 5.8. A T space X is called a countable - directed closure space , or ω ∗ - DC space for short, if Irr c ( X ) = D ω ∗ c ( X ), that is, for each A ∈ Irr c ( X ), there exists a countably-directed subset D of X such that A = D .Now we introduce another type of weak well-filtered spaces. Definition 5.9. A T space X is called ω ∗ - well - filtered , if for any countable-filtered family { K i : i ∈ I } ⊆ K ( X ) (that is, { K i : i ∈ I } ∈ D ω ∗ ( K ( X ))) and U ∈ O ( X ), it satisfies \ i ∈ I K i ⊆ U ⇒ ∃ j ∈ I, K j ⊆ U. Clearly, every well-filtered space is ω ∗ -well-filtered. The converse implications does not hold in general,as shown by the following example. Example 5.10.
Let P = ( N ( <ω ) , ⊆ ) and X = Σ ω ∗ P . It is easy to verify that any countably-directed subsetof P has a largest element. Therefore, σ ω ∗ ( P ) = α ( P ) and K ( X ) = {↑ F : F ∈ N ( <ω ) } , and hence X is ω ∗ -well-filtered as any countable-filtered family { K i : i ∈ I } ⊆ K ( X ) has a least element. Since P is not adcpo ( P is directed but has not a largest element), X is not a d -space, and hence not well-filtered. Proposition 5.11.
Every ω ∗ -well-filtered space is an ω ∗ - d -space. Proof . Let X be an ω ∗ -well-filtered space and D ∈ D ω ∗ ( X ). Then {↑ d : d ∈ D } ∈ D ω ∗ ( K ( X )). By the ω ∗ -well-filteredness of X , we have T d ∈ D ↑ d * X \ D or, equivalently, T d ∈ D ↑ d ∩ D = ∅ . Therefore, there is x ∈ T d ∈ D ↑ d ∩ D , and hence D = { x } .In the following, using the topological Rudin’s Lemma, we prove that a T space X is ω ∗ -well-filterediff the Smyth power space of X is ω ∗ -well-filtered iff the Smyth power space of X is ω ∗ - d -space. Thecorresponding results for well-filteredness and ω -well-filteredness are proved in [19, 20, 21]. Theorem 5.12.
For a T space X , the following conditions are equivalent: (1) X is ω ∗ -well-filtered. (2) P S ( X ) is an ω ∗ - d -space. (3) P S ( X ) is ω ∗ -well-filtered. Proof . (1) ⇒ (2): Suppose that X is an ω ∗ -well-filtered space. For any countable-filtered family K ⊆ K ( X ),by the ω ∗ -well-filteredness of X , T K ∈ K ( X ). Therefore, by Lemma 2.6, K ( X ) is an ω ∗ -dcpo. Clearly, bythe ω ∗ -well-filteredness of X , U ∈ σ ω ∗ ( K ( X )) for any U ∈ O ( X ). Thus P S ( X ) is an ω ∗ - d -space.(2) ⇒ (3): Suppose that {K i : i ∈ I } ⊆ K ( P S ( X )) is countable-filtered, U ∈ O ( P S ( X )), and T i ∈ I K i ⊆ U .If K i
6⊆ U for all i ∈ I , then by Lemma 3.1, K ( X ) \ U contains an irreducible closed subset A that stillmeets all K i ( i ∈ I ). For each i ∈ I , let K i = S ↑ K ( X ) ( A T K i ) (= S ( A T K i )). Then by Lemma 2.7, { K i : i ∈ I } ⊆ K ( X ) is countable-filtered, and K i ∈ A for all i ∈ I since A = ↓ K ( X ) A . Let K = T i ∈ I K i . Then K ∈ K ( X ) and K = W K ( X ) { K i : i ∈ I } ∈ A by Lemma 2.6 and condition (2). We claim that K ∈ T i ∈ I K i .Suppose, on the contrary, that K T i ∈ I K i . Then there is a j ∈ I such that K
6∈ K j . Select a G ∈ A T K j .Then K G (otherwise, K ∈ ↑ K ( X ) K j = K j , being a contradiction with K
6∈ K j ), and hence there is a g ∈ K \ G . It follows that g ∈ K i = S ( A T K i ) for all i ∈ I and G K ( K ) { g } . For each i ∈ I , by g ∈ K i = S ( A T K i ), there is a K gi ∈ A T K i such that g ∈ K gi , and hence K gi ∈ K ( K ) { g } T A T K i .Thus K ( K ) { g } T A T K i = ∅ for all i ∈ I . By the minimality of A , we have A = K ( K ) { g } T A , and7onsequently, G ∈ A T K j = K ( K ) { g } T A T K j , which is a contradiction with G K ( K ) { g } . Thus K ∈ T i ∈ I K i ⊆ U ⊆ K ( X ) \ A , being a contradiction with K ∈ A . Therefore, P S ( X ) is ω ∗ -well-filtered.(3) ⇒ (1): Suppose that K ⊆ K ( X ) is countable-filtered, U ∈ O ( X ), and T K ⊆ U . Let e K = {↑ K ( X ) K : K ∈ K} . Then e K ⊆ K ( P S ( X )) is countable-filtered and T e
K ⊆ U . By the ω ∗ -well-filteredness of P S ( X ),there is a K ∈ K such that ↑ K ( X ) K ⊆ U , and whence K ⊆ U , proving that X is ω ∗ -well-filtered. Definition 5.13. A T space X is called a countably-directed closure space, ω ∗ - DC space for short, if Irr c ( X ) = D ω ∗ c ( X ), that is, for each A ∈ Irr c ( X ), there exists a countably-directed subset of X such that A = D .By Remark 2.6, Proposition 5.7 and Proposition 5.11, we get the following result. Proposition 5.14.
For any T space X , the following conditions are equivalent: (1) X is sober. (2) X is an ω ∗ - DC and ω ∗ -well-filtered space. (3) X is an ω ∗ - DC and ω ∗ - d -space.
6. First-countability of sobrifications and ω -Rudin spaces In this section, we prove that if the sobrification of a T space X is first-countable, then X is a ω -Rudinspace. Hence every ω -well-filtered space having a first-countable sobrification is sober.We first prove two useful lemmas. Lemma 6.1.
Let X be a T space and A ∈ C ( X ) . For { U n : n ∈ N } ⊆ O ( X ) with U ⊇ U ⊇ ... ⊇ U n ⊇ U n +1 ⊇ ... , if A ∈ m ( { U n : n ∈ N } and x n ∈ U n ∩ A for each n ∈ N , then every subset of { x n : n ∈ N } ) iscompact. Proof . Suppose E ⊆ { x n : n ∈ N } and { V i : i ∈ I } is an open cover of E , that is, E ⊆ S i ∈ I V i .Case 1. E ∩ ( X \ V j ) is finite for some j ∈ I .Then there is I j ∈ I ( <ω ) such that E ∩ ( X \ V j ) ⊆ S i ∈ I j V i , and hence E ⊆ V j ∪ S i ∈ I j V i .Case 2. E ∩ ( X \ V i ) is infinite for all i ∈ I .For each i ∈ I , since U ⊇ U ⊇ ... ⊇ U n ⊇ U n +1 ⊇ ... and x n ∈ U n ∩ A for each n ∈ N , we have that U n ∩ A ∩ ( X \ V i ) = ∅ for all n ∈ N , and hence A ∩ ( X \ V i ) = A by the minimality of A . It follows that A ⊆ T i ∈ I ( X \ V i ) = X \ S i ∈ I V i . Therefore, E ⊆ A ∩ S i ∈ I V i = ∅ .By Case 1 and Case 2, E is compact. Lemma 6.2.
Let X be a T space and A ∈ Irr c ( X ) . For any open neighborhood base { U i : i ∈ I } of A in X s , A ∈ m ( { U i : i ∈ I } ) . Proof . Clearly, A ∈ M ( { U i : i ∈ I } ). Suppose B ∈ C ( X ) and B ⊆ A . If B = A , then A ∩ ( X \ B ) = ∅ ,and hence A ∈ ( X \ B ). Since { U i : i ∈ I } is an open neighborhood base at A in X s , there is j ∈ I suchthat U j ⊆ ( X \ B ) or, equivalently, U j ⊆ X \ B . Therefore, U j ∩ B = ∅ . So B / ∈ M ( { U i : i ∈ I } ). Thus A ∈ m ( { U i : i ∈ I } ). Proposition 6.3.
For a T space X , the following two conditions are equivalent: (1) X is second-countable. (2) X s is second-countable. Proof . For any { U i : i ∈ I } ⊆ O ( X ), it is easy to verify that { U i : i ∈ I } is a base of X iff { U i : i ∈ I } isa base of X s .Since the first-countability is a hereditary property, by Remark 2.5, we have the following result.8 roposition 6.4. For a T space X , if X s is first-countable, then X is first-countable. The converse of Proposition 6.4 does not hold in general, as shown in Example 6.13 below.
Proposition 6.5.
If a T space X is first-countable and | X | ≤ ω , then X s is first-countable. Proof . Let X = { x n : n ∈ N } . For each n ∈ N , since X is first-countable, there is a countable base { U m ( x n ) : m ∈ N } at x n . For any A ∈ Irr c ( X ), it straightforward to verify that { U m ( x n ) : ( m, n ) ∈ N × N and A ∩ U m ( x n ) = ∅} is a countable base at A in X s . Thus X s is first-countable.The following example shows that the Scott topology on a countable complete lattice may not be first-countable. Example 6.6. ([20])
Let L = {⊥} ∪ ( N × N ) ∪ {⊤} and define a partial order ≤ on L as follows:(i) ∀ ( n, m ) ∈ N × N , ⊥ ≤ ( n, m ) ≤ ⊤ ;(ii) ∀ ( n , m ) , ( n , m ) ∈ N × N , ( n , m ) ≤ ( n , m ) iff n = n and m ≤ m .Then ( L, σ ( L )) does not have any countable base at ⊤ . Assume, on the contrary, there exists a countablebase { U n : n ∈ N } at ⊤ . Then for each n ∈ N , as _ ( { n } × N ) = ⊤ ∈ U n , there exists m n ∈ N such that ( n, m n ) ∈ U n . Let U = S n ∈ N ↑ ( n, m n + 1). Then U ∈ σ ( L ). But for each n ∈ N , ( n, m n ) ∈ U n \ U , which contradicts that { U n : n ∈ N } is a base at ⊤ . Therefore, ( L, σ ( L )) is notfirst-countable. One can easily check that ( L, σ ( L )) is sober. Theorem 6.7.
For a T space X , if X s is first-countable, then X is an ω -Rudin space. Proof . Let A ∈ Irr c ( X ). By the first-countability of X s , there is an open neighborhood base { U n : n ∈ N } of A such that U ⊇ U ⊇ . . . ⊇ U n ⊇ . . . , or equivalently, U ⊇ U ⊇ . . . ⊇ U n ⊇ . . . . By Lemma 6.2, A ∈ m ( { U n : n ∈ N } . For each n ∈ N , choose an x n ∈ U n ∩ A , and let K n = ↑{ x m : m ≥ n } . Then K ⊇ K ⊇ . . . ⊇ K n ⊇ . . . , and { K n : n ∈ N } ⊆ K ( X )by Lemma 6.1. Clearly, A ∈ M ( { K n : n ∈ N } ). For any B ∈ C ( X ), if B is a proper subset of A , that is, A ∩ ( X \ B ) = A \ B = ∅ , then A ∈ ( X \ B ) ∈ O ( X s ) (= O ( P H ( Irr c ( X )))). Therefore, U m ⊆ ( X \ B )for some m ∈ N , and hence U m ⊆ X \ B or, equivalently, U m ∩ B = ∅ . Thus B / ∈ M ( { K n : n ∈ N } ), proving A ∈ m ( { K n : n ∈ N } ). So X is an ω -Rudin space. Corollary 6.8.
Every second-countable T space is an ω -Rudin space. Corollary 6.9.
Every countable first-countable T space is an ω -Rudin space. Theorem 6.10.
Every ω -well-filtered space with a first-countable sobrification is sober Proof . For A ∈ Irr c ( X ), by Theorem 6.7 and its proof (or Proposition 4.6), there is a decreasing sequence { K n : n ∈ N } ⊆ K ( X ) such that A ∈ m ( { K n n ∈ N } . Since X is ω -well-filtered, T n ∈ N K n * X \ A , that is, T n ∈ N K n ∩ A = ∅ . Choose x ∈ T n ∈ N K n ∩ A . Then { x } ∈ M ( { K n n ∈ N } and { x } ⊆ A . By the minimalityof A , we have A = { x } . Thus X is sober. Corollary 6.11.
Every second-countable ω -well-filtered space is sober. Corollary 6.12.
Every countable first-countable ω -well-filtered space is sober. In Theorem 6.7 and Theorem 6.10, the first-countability of X s can not be weakened to that of X asshown in the following example (see also Example 7.5 in Section 7).9 xample 6.13. Let ω be the first uncountable ordinal number and P = [0 , ω ). Then(a) C (Σ P ) = {↓ t : t ∈ P } ∪ {∅ , P } .(b) Σ P is compact since P has a least element 0.(c) Σ P is first-countable.(d) (Σ P ) s is not first-countable. In fact, it is easy to verify that (Σ P ) s is homeomorphic to Σ[0 , ω ]. Sincesup of a countable family of countable ordinal numbers is still a countable ordinal number, Σ[0 , ω ] hasno countable base at the point ω .(e) P is an ω -dcpo but not a dcpo. So Σ P is an ω - d -space but not a d -space, and hence not a sober space.(f) K (Σ P ) = {↑ x : x ∈ P } . For K ∈ K (Σ P ), we have inf K ∈ K , and hence K = ↑ inf K .(g) Σ P is a Rudin space. One can easily check that Irr c (Σ P ) = {↓ x : x ∈ P } ∪ { P } . Clearly, ↓ x is aRudin set for each x ∈ P . Now we show that P is a Rudin set. First, {↑ s : s ∈ P } is filtered. Second, P ∈ M ( {↑ s : s ∈ P } ). For a closed subset B of Σ P , if B = P , then B = ↓ t for some t ∈ P , and hence ↑ ( t + 1) ∩ ↓ t = ∅ . Thus B / ∈ M ( {↑ s : s ∈ P } ), proving that P is a Rudin set.(h) Σ P is not an ω -Rudin space. We prove that the irreducible closed set P is not an ω -Rudin set. For anycountable filtered family {↑ α n : n ∈ N } ⊆ K (Σ P ), let β = sup { α n : n ∈ N } . Then β is still a countableordinal number. Clearly, ↓ β ∈ M ( {↑ α n : n ∈ N } and P = ↓ β . Therefore, P / ∈ m ( {↑ α n : n ∈ N } ). Thus P is not an ω -Rudin set, and hence Σ P is not an ω -Rudin space.(i) Σ P is ω -well-filtered. If {↑ x n : n ∈ N } ⊆ K (Σ P ) is countable filtered family and U ∈ σ ( P ) with T n ∈ N ↑ x n ⊆ U , then { x n : i ∈ N } is a countable subset of P = [0 , ω ). Since sup of a countable familyof countable ordinal numbers is still a countable ordinal number, we have β = sup { x n : n ∈ N } ∈ P ,and hence ↑ β = T n ∈ N ↑ x n ⊆ U . Therefore, β ∈ U , and consequently, x n ∈ U for some n ∈ N or,equivalently, ↑ x n ⊆ U , proving that Σ P is ω -well-filtered.
7. First-countability and well-filtered determined spaces
In this section, we show that any first-countable T space is well-filtered determined. In [20] it wasshown that in a first-countable ω -well-filtered T space X , all irreducible closed subsets of X are directed(see [20, Theorem 4.1]). In the following we will strengthen this result by proving that in a first-countable ω -well-filtered space X , every irreducible closed subset of X is countably-directed. Lemma 7.1.
Suppose that X is a first-countable T space, Y is an ω -well-filtered space and f : X → Y isa continuous mapping. Then for any A ∈ Irr ( X ) and { a n : n ∈ N } ⊆ A , T n ∈ N ↑ f ( a n ) T f ( A ) = ∅ . Proof . For each x ∈ X , since X is first-countable, there is an open neighborhood base { U n ( x ) : n ∈ N } of x such that U ( x ) ⊇ U ( x ) ⊇ . . . ⊇ U k ( x ) ⊇ . . . , that is, { U n ( x ) : n ∈ N } is a decreasing sequence of open subsets.For each ( n, m ) ∈ N × N , since a n ∈ A and A ∈ Irr ( X ), we have A ∩ m T i =1 U l i ( a k i ) ∩ A = ∅ for all { ( l i , k i ) ∈ N × N : 1 ≤ i ≤ m } .Choose c ∈ U ( a ) ∩ ∩ A . Now suppose we already have a set { c , c , . . . , c n − } such that for each2 ≤ i ≤ n − c i ∈ i − \ j =1 U i ( c j ) ∩ i \ j =1 U i ( a j ) ∩ A. Note that above condition implies that for any positive integer 1 ≤ k ≤ n − k \ j =1 U k ( c j ) ∩ k \ j =1 U i ( a j ) ∩ A = ∅ , and10 − \ j =1 U n ( c j ) ∩ n \ j =1 U n ( a j ) ∩ A = ∅ . So we can choose c n ∈ n − T j =1 U n ( c j ) ∩ n T j =1 U n ( a j ) ∩ A . By A ∈ Irr ( X ) again, we have n \ j =1 U n ( c j ) ∩ n \ j =1 U n ( a j ) ∩ A = ∅ . By induction, we can obtain a set { c n : n ∈ N } .Let K n = ↑{ c k : k ≥ n } for each n ∈ N . Claim 1: ∀ n ∈ N , K n ∈ K ( X ).Suppose { V i : i ∈ I } is an open cover of K n , i.e., K n ⊆ S i ∈ I V i . Then there is i ∈ I such that c n ∈ V i ,and thus there is m ≥ n such that c n ∈ U m ( c n ) ⊆ V i . It follows that c k ∈ U m ( c n ) ⊆ V i for all k ≥ m .Thus { c k : k ≥ m } ⊆ V i . For each c k , where n ≤ k < m , choose a V i k such that c k ∈ V i k . Then the finitefamily { V i k : n ≤ k < m } ∪ { V i } covers K n . So K n is compact. Claim 2: {↑ f ( K n ) : n ∈ N } ⊆ K ( Y ) and ↑ f ( K ) ⊇ ↑ f ( K ) ⊇ ... ⊇ ↑ f ( K n ) ⊇ ↑ f ( K n +1 ) ⊇ ... For each n ∈ N , since K m ∈ K ( X ) and f is continuous, we have ↑ f ( K m ) ∈ K ( Y ). Clearly, K ⊇ K ⊇ ... ⊇ K n ⊇ K n +1 ⊇ ... , and hence ↑ f ( K ) ⊇ ↑ f ( K ) ⊇ ... ⊇ ↑ f ( K n ) ⊇ ↑ f ( K n +1 ) ⊇ ... Claim 3: T n ∈ N ↑ f ( K n ) = T n ∈ N ↑ f ( c n ).Clearly, T n ∈ N ↑ f ( c n ) ⊆ T n ∈ N ↑ f ( K n ). Now we show T n ∈ N ↑ f ( K n ) ⊆ ↑ f ( c m ) for all m ∈ N . Suppose V ∈O ( Y ) with f ( c m ) ∈ V . Then c m ∈ f − ( V ) ∈ O ( X ), and whence U n ( m ) ( c m ) ⊆ f − ( V ) for some n ( m ) ∈ N .For any l ≥ max { m, n ( m ) } +1, we have K l ⊆ U l ( c m ) ⊆ U n ( m ) ( c m ) ⊆ f − ( V ), and consequently, ↑ f ( K l ) ⊆ V .It follows that T n ∈ N ↑ f ( K n ) ⊆ T f ( c m ) ∈ V ∈O ( Y ) V = ↑ f ( c m ). Thus T n ∈ N ↑ f ( K n ) ⊆ T n ∈ N ↑ f ( c n ). Claim 4: T n ∈ N ↑ f ( K n ) ⊆ T n ∈ N ↑ f ( a n ).For m ∈ N and W ∈ O ( Y ) with f ( a m ) ∈ W . Then a m ∈ f − ( W ∈ O ( X ), and whence U k ( m ) ( a m ) ⊆ f − ( W ) for some k ( m ) ∈ N . For any l ≥ max { m, k ( m ) } , we have K l ⊆ U l ( a m ) ⊆ U k ( m ) ( a m ) ⊆ f − ( W ),and consequently, ↑ f ( K l ) ⊆ W . It follows that T n ∈ N ↑ f ( K n ) ⊆ T f ( a m ) ∈ W ∈O ( Y ) W = ↑ f ( a m ). Thus T n ∈ N ↑ f ( K n ) ⊆ T n ∈ N ↑ f ( a n ). Claim 5: T n ∈ N ↑ f ( K n ) ∈ K ( Y ) and T n ∈ N ↑ f ( K n ) ∩ f ( A ) = ∅ .By Claim 2 and the ω -well-filteredness of Y , T n ∈ N ↑ f ( K n ) ∈ K ( Y ). Now we show T n ∈ N ↑ f ( K n ) ∩ f ( A ) = ∅ . Assume, on the contrary, that T n ∈ N ↑ f ( K n ) ∩ f ( A ) = ∅ or, equivalently, T n ∈ N ↑ f ( K n ) ⊆ Y \ f ( A ).Then by the ω -well-filteredness of Y and Claim 2, ↑ f ( K n ) ⊆ Y \ f ( A ), which is in contradiction with { c m : m ≥ n } ⊆ A ∩ K n . Therefore, T n ∈ N ↑ f ( K n ) ∩ f ( A ) = ∅ . Corollary 7.2.
In a first-countable ω -well-filtered space X , every irreducible closed subset of X is countably-directed. Therefore, X is an ω ∗ - DC space. By Remark 2.8, Proposition 5.7 and Corollary 7.2, we get the following result.
Theorem 7.3.
For a first-countable T space X , the following conditions are equivalent: (1) X is a sober space. (2) X is a well-filtered space. (3) X is an ω -well-filtered d -space. (4) X is an ω -well-filtered ω ∗ - d -space. A first-countable d -space may not be sober as shown in the following example. Example 7.4.
Let X be a countably infinite set and X cof the space equipped with the co-finite topology (the empty set and the complements of finite subsets of X are open). Then(a) C ( X cof ) = {∅ , X } ∪ X ( <ω ) , X cof is T and hence a d -space.11b) K ( X cof ) = 2 X \ {∅} .(c) X cof is first-countable.(d) X cof is locally compact and hence a Rudin space by Theorem 3.9.(e) X cof is non-sober. K X = { X \ F : F ∈ X ( <ω ) } ⊆ K ( X cof ) is countable filtered and T K X = X \ S X ( <ω ) = X \ X = ∅ , but X \ F = ∅ for all F ∈ X ( <ω ) . Thus X cof is not ω -well-filtered, andconsequently, X cof is not sober by Theorem 7.4.The following example shows that a first-countable ω -well-filtered space need not to be sober. Example 7.5.
Let L be the complete chain [0 , ω ]. Then(a) Σ ω L is a first-countable.(b) σ ( P ) = σ ω ( L ). Since sups of all countable families of countable ordinal numbers are still countableordinal numbers, we have that { ω } ∈ σ ω ( L ) but { ω } / ∈ σ ( L ) (note that ω = sup [0 , ω )).(c) σ ( L ) = σ ω ∗ ( L ). It is easy to check that [ ω, ω ] ∈ σ ω ∗ ( L ) but [ ω, ω ] / ∈ σ ( L ) (note that ω = sup N ).(d) K (Σ ω L ) = {↑ α : α ∈ [0 , ω ] } . For K ∈ K , we have inf K ∈ K , and hence K = ↑ inf K .(e) Σ ω L is not an ω -Rudin space. It is easy to check that [0 , ω ) ∈ Irr c (Σ ω L ) (note that { ω } ∈ σ ω ( L )).If [0 , ω ) ∈ RD ω (Σ ω L ), then by (d), there is a countable subset { α n : n ∈ N } ⊆ [0 , ω ) such that[0 , ω ) ∈ m ( {↑ α n : n ∈ N } ). Let β = sup { α n : n ∈ N } . Then β ∈ [0 , ω ), and hence ↓ β ∈ C (Σ ω L ) and ↓ β ∈ M ( {↑ α n : n ∈ N } ), which is in contradiction with [0 , ω ) ∈ m ( {↑ α n : n ∈ N } ).(f) Σ ω L is ω -well-filtered. Suppose that {↑ α n : n ∈ N } ⊆ K (Σ ω L ) is countable filtered and U ∈ σ ω ( L ) with T n ∈ N ↑ α n ⊆ U . Let α = sup { α n : n ∈ N } . Then { α n : n ∈ N } is a countable directed subset of L and α ∈ U since ↑ α = T n ∈ N ↑ α n ⊆ U . It follows that α n ∈ U or, equivalently, ↑ α n ⊆ U for some n ∈ N .Thus Σ ω L is ω -well-filtered, and hence an ω - d -space.(g) Σ ω L is not well-filtered. {↑ t : t ∈ [0 , ω ) } ⊆ K (Σ ω L ) is filtered and T t ∈ [0 ,ω ) ↑ t = { ω } ∈ σ ω ( L ), but ↑ t * { ω } for all t ∈ [0 , ω ). Therefore, Σ ω L is not well-filtered.(h) Σ ω L is not a d -space. [0 , ω ) ∈ Irr c (Σ ω L ) and [0 , ω ) is directed, but [0 , ω ) = cl σ ω ( L ) { α } = [0 , α ] for all α ∈ L . Thus Σ ω L is not a d -space.(i) Σ ω L is not an ω ∗ - d -space. In fact, by (b), { ω } ∈ σ ω ( L ) but { ω } / ∈ σ ω ∗ ( L ), and hence by Theorem5.7, Σ ω L is not an ω ∗ - d -space.Since Σ ω L is not well-filtered, it is non-sober. So in Theorem 7.3, condition (4) (and so condition (3))cannot be weakened to the condition that X is only an ω -well-filtered space. Definition 7.6.
Let X be a first-countable T space, A ∈ Irr ( X ) and { a n : n ∈ N } . The countable family { K n : n ∈ N } ⊆ K ( X ) obtained in the proof of Lemma 7.1 is called a decreasing sequence of compactsaturated subsets related to { a n : n ∈ N } . Theorem 7.7.
Suppose that X is a first-countable T space, Y is an ω -well-filtered space and f : X → Y is a continuous mapping. Then for any A ∈ Irr ( X ) , f ( A ) ∈ RD ( Y ) . Proof . Let K A = { T n ∈ N ↑ f ( K n ) : { K n : n ∈ N } is a decreasing sequence of compact saturated subsetsrelated to a countable set { a n : n ∈ N } ⊆ A } . Then by the proof of Lemma 7.1, we have ◦ K A = ∅ and K A ⊆ K ( Y ). ◦ ↑ f ( a ) ∈ K A for all a ∈ A .For { a n : n ∈ N } ⊆ A with a n ≡ a , as carrying out in the proof of Lemma 7.1, choose c n ≡ a for all n ∈ N . Then K = K = ... = K n = ... = ↑ a , and hence ↑ f ( a ) = ↑ f ( ↑ a ) = T n ∈ N ↑ f ( K n ) ∈ K A . ◦ K A is filtered.Suppose that { K n : n ∈ N } and { G n : n ∈ N } are decreasing sequences of compact saturated subsetsrelated to countable sets { a n : n ∈ N } ⊆ A and { b n : n ∈ N } ⊆ A , respectively. By the proof of Lemma7.1, for each n ∈ N , K n = ↑{ c m : m ≥ n } and G n = ↑{ d m : m ≥ n } , where { c n : n ∈ N } ⊆ A and { d n : n ∈ N } ⊆ A are obtained by the choice procedures (by induction) in the proof of Lemma 7.1 related to12 a n : n ∈ N } ⊆ A and { b n : n ∈ N } ⊆ A , respectively. Consider { s n : n ∈ N } = { c , d , c , d , ..., c n , d n , ... } ⊆ A , that is, s n = ( c k n = 2 k + 1 d k n = 2 k. Then by the proof of Lemma 7.1, we can inductively choose a countable set { t n : n ∈ N } such that t n ∈ n − \ j =1 U n ( t j ) ∩ n \ j =1 U n ( s j ) ∩ A forall n ∈ N . For each n ∈ N , let H n = ↑{ t m : m ≥ n } . Then by Claim 1 and Claim 2 in the proof of Lemma 7.1, ↑ f ( H n ) ∈ K A . By Claim 3 and Claim 4 in the proof of Lemma 7.1, we have T n ∈ N ↑ f ( H n ) = T n ∈ N ↑ f ( t n ) ⊆ T n ∈ N ↑ f ( s n ) = T n ∈ N ↑ f ( c n ) ∩ T n ∈ N ↑ f ( d n ) = T n ∈ N ↑ f ( K n ) ∩ T n ∈ N ↑ f ( G n ). Thus K A is filtered. ◦ f ( A ) ∈ M ( K A ).By Claim 5 in the proof of Lemma 7.1, f ( A ) ∈ M ( K A ). ◦ f ( A ) ∈ m ( K A ).If B is a closed subset with B ∈ M ( K A ), then for each a ∈ A , by 2 ◦ , we have ↑ f ( a ) ∩ B = ∅ , and hence f ( a ) ∈ B . It follows that f ( A ) ⊆ B . Thus f ( A ) ∈ m ( K A ).By 1 ◦ , 3 ◦ and 5 ◦ , f ( A ) ∈ RD ( Y ). Theorem 7.8.
Every first-countable T space is a well-filtered determined space. Proof . Let X be a first-countable T space and A ∈ Irr c ( X ). We need to show A ∈ WD ( X ). Suppose that f : X −→ Y is a continuous mapping from X to a well-filtered space Y . By Theorem 7.7, f ( A ) ∈ RD ( Y ), andwhence by the well-filteredness of Y and Proposition 3.6, there is a (unique) y A ∈ Y such that f ( A ) = { y A } .Thus A ∈ WD ( X ).In [14, Example 4.15], a well-filtered space X but not a Rudin space was given. It is straightforward tocheck that X is not first-countable.Finally, based on Theorem 6.7 and Theorem 7.8, we pose the following two natural problems. Problem 7.9.
Is every first-countable T space a Rudin space? Problem 7.10.
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