On open well-filtered spaces
LLogical Methods in Computer ScienceVolume 16, Issue 4, 2020, pp. 18:1–18:10https://lmcs.episciences.org/ Submitted Jan. 22, 2020Published Dec. 18, 2020
ON OPEN WELL-FILTERED SPACES
CHONG SHEN, XIAOYONG XI, XIAOQUAN XU, AND DONGSHENG ZHAOSchool of Mathematical Sciences, Nanjing Normal University, Jiangsu, Nanjing, China e-mail address : [email protected] of Mathematics and Statistics, Jiangsu Normal University, Jiangsu, Xuzhou, China e-mail address : [email protected] of Mathematics and Statistics, Minnan Normal University, Fujian, Zhangzhou, China e-mail address : [email protected] and Mathematics Education, National Institute of Education, Nanyang TechnologicalUniversity, 1 Nanyang Walk, Singapore e-mail address : [email protected]
Abstract.
We introduce and study a new class of T spaces, called open well-filteredspaces. The main results we prove include (i) every well-filtered space is an open well-filteredspace; (ii) every core-compact open well-filtered space is sober. As an immediate corollary,it follows that every core-compact well-filtered space is sober. This provides a different andrelatively more straightforward method to answer the problem posed by Jia and Jung: isevery core-compact well-filtered space sober? Introduction
The sobriety is one of the most important topological properties, particularly meaningfulfor T spaces. It has been used in the characterization of spectral spaces of commutativerings and the spaces which are determined by their open set lattices. In domain theory,it was proved that the Scott space of every domain is sober quite early on. Since thenthe investigation of the sobriety of Scott spaces of general directed complete posets led tomany deep results. Heckmann introduced the well-filtered spaces and asked whether everywell-filtered Scott space of a directed complete poset is sober [Hec90, Hec91]. This questioninspired intensive studies on the relationship between sobriety and well-filteredness (see[HGJX18, JJL16, Kou01, ZXC19, XZ17, XL17, WXXZ19]). A recent problem on this topic Key words and phrases: well-filtered space, ω -well-filtered space, sober space, core-compact space, locallycompact space, open well-filtered space.The first author is sponsored by NSFC (11871097) and the China Scolarship Council (File No.201806030073).The second author is sponsored by NSFC (1207188).The third author is sponsored by NSFC (11661057, 1207199), the Ganpo 555 project for leading talent ofJiangxi Provence and the Natural Science Foundation of Jiangxi Province, China (20192ACBL20045). LOGICAL METHODS l IN COMPUTER SCIENCE DOI:10.23638/LMCS-16(4:18)2020 © C. Shen, X. Xi, X. Xu, and D. Zhao CC (cid:13) Creative Commons
C. Shen, X. Xi, X. Xu, and D. Zhao
Vol. 16:4 is whether every core-compact well-filtered space is sober, posed by Jia and Jung [Jia18].The problem has been answered positively by Lawson, Wu and Xi [LWX20] and Xu, Shen,Xi and Zhao [XSXZ20].In the current paper we first introduce a new class of topological spaces, called openwell-filtered spaces, which includes all well-filtered spaces. The open well-filtered spacesthemselves may deserve further study that will enrich the theory of T topological spaces.We prove that (i) every well-filtered space is an open well-filtered space, and (ii) everycore-compact open well-filtered space is sober. As an immediate implication, we obtain thatevery core-compact well-filtered space is sober, thus giving a relatively more straightforwardmethod to answer Jia and Jung’s problem [Jia18].2. Preliminaries
This section is devoted to a brief review of some basic concepts and notations that will beused in the paper. For more details, see [Eng89, GHK +
03, GL13].Let P be a poset. A nonempty subset D of P is directed if every two elements in D have an upper bound in D . P is called a directed complete poset , or dcpo for short, if forany directed subset D ⊆ P , (cid:87) D exists.Let X be a T space. A subset A of X is called saturated if A equals the intersectionof all open sets containing it. The specialization order ≤ on X is defined by x ≤ y iff x ∈ cl( { y } ), where cl is the closure operator. It is easy to show that a subset A of X issaturated if and only if A = ↑ A = { x ∈ X : x ≥ a for some a ∈ A } with respect to thespecialization order.A nonempty subset A of X is irreducible if for any closed sets F , F of X , A ⊆ F ∪ F implies A ⊆ F or A ⊆ F . A T space X is called sober if for any irreducible closed set F , F = ↓ x = cl( { x } ) for some x ∈ X .For a T space X , we shall consider the following subfamilies of the power set 2 X : Q ( X ) , the set of all compact saturated subsets of X ; S ( X ) , the set of all saturated subsets of X ; O ( X ) , the set of all open subsets of X. For
A, B ⊆ X , we say that A is relatively compact in B , denoted by A (cid:28) B , if A ⊆ B and every open cover of B contains a finite subcover of A .We write A ⊆ flt X ( Q ( X ) , S ( X ) , O ( X ) , resp.)for that A is a (cid:28) -filtered subfamily of 2 X ( Q ( X ), S ( X ), O ( X ), resp.), that is, ∀ A , A ∈ A ,there exists A ∈ A such that A (cid:28) A , A . Remark 2.1. (1) In general, for any
A, B ⊆ X , that each open cover of B contains a finitesubcover of A does not imply A ⊆ B . For example, on the set X = { x, y } , consider thetopology O ( X ) = {∅ , { x } , X } . Then X is a T space and { x } (cid:28) { y } . Thus the requirement A ⊆ B in the definition of A (cid:28) B is not redundant.(2) For any A ⊆ X and B ∈ O ( X ), A (cid:28) B if and only if each open cover of B containsa finite subcover of A . This is because { B } is an open cover of A .(3) For any A ⊆ X and Q ∈ Q ( X ), A (cid:28) Q if and only if A ⊆ Q . Hence, A ⊆ flt Q ( X )if and only if ( A , ⊇ ) is a directed family.A T space X is called well-filtered if for any K ⊆ flt Q ( X ) and U ∈ O ( X ), (cid:84) K ⊆ U implies K ⊆ U for some K ∈ K . We note that every sober space is well-filtered [GHK + ol. 16:4 OPEN WELL-FILTERED SPACES 18:3 In what follows, the symbol ω will denote the smallest infinite ordinal, and for any set X , the family of all finite subsets of X is denoted by X ( <ω ) . Definition 2.2. A T space X is called ω -well-filtered , if for any { K n : n < ω } ⊆ flt Q ( X )and U ∈ O ( X ), (cid:92) n<ω K n ⊆ U ⇒ ∃ n < ω, K n ⊆ U. Proposition 2.3. A T space X is ω -well-filtered if and only if for any descending chain { K n : n < ω } ⊆ Q ( X ) , that is, K ⊇ K ⊇ K ⊇ . . . ⊇ K n ⊇ K n +1 ⊇ . . . , and U ∈ O ( X ) , (cid:92) n<ω K n ⊆ U ⇒ ∃ n < ω, K n ⊆ U. Proof.
We only need to prove the sufficiency. Let
K ⊆ flt Q ( X ) be a countable family and U ∈ O ( X ) such that (cid:84) K ⊆ U .If the cardinality |K| < ω , i.e., K is a finite family, then K contains a smallest element Q , and hence Q = (cid:84) K ⊆ U , completing the proof.Now assume |K| = ω . We may let K = { K n : n < ω } . We use induction on n < ω todefine a descending chain (cid:98) K = (cid:110) (cid:98) K n : n < ω (cid:111) . Specifically, let (cid:98) K = K and let (cid:98) K n +1 ∈ K be a lower bound of (cid:110) K n +1 , (cid:98) K , (cid:98) K , (cid:98) K . . . , (cid:98) K n (cid:111) under the inclusion order. Then (cid:98) K ⊆ K isa descending chain and (cid:98) K n ⊆ K n for all n < ω , implying that (cid:84) (cid:98) K = (cid:84) K ⊆ U . Then byassumption, there exists n < ω such that (cid:98) K n ⊆ U , completing the proof. Lemma 2.4.
Let X be a T space and A ⊆ flt X . Each closed set C ⊆ X that intersectsall members of A contains a minimal (irreducible) closed subset F that still intersects allmembers of A .Proof. Let B := { B ∈ C ( X ) : ∀ A ∈ A , B ∩ A (cid:54) = ∅} , where C ( X ) is the set of all closed subsetsof X .(i) B (cid:54) = ∅ because C ∈ B .(ii) Let H ⊆ B be a chain. We claim that (cid:84)
H ∈ B . Otherwise, there exists A ∈ A such that A ∩ (cid:84) H = ∅ . As A is (cid:28) -filtered, there exists A ∈ A such that A (cid:28) A . Since { X \ B : B ∈ H} is a directed open cover of A , there exists B ∈ H such that A ⊆ X \ B implying A ∩ B = ∅ . This means that B / ∈ B , contradicting B ∈ H ⊆ B .By Zorn’s Lemma, it turns out that there exists a minimal closed subset F ⊆ C suchthat F ∩ A (cid:54) = ∅ for all A ∈ A .Now we show that F is irreducible. If F is not irreducible, then there exist closed sets F , F such that F = F ∪ F but F (cid:54) = F and F (cid:54) = F . Since F , F are proper subsets of F ,by the minimality of F , there exist A , A ∈ A such that F ∩ A = ∅ and F ∩ A = ∅ . Since A is a (cid:28) -filtered family, there exists A ∈ A such that A (cid:28) A , A (hence A ⊆ A , A ).It then follows that F ∩ A ⊆ F ∩ A = ∅ and F ∩ A ⊆ F ∩ A = ∅ , which impliesthat F ∩ A = F ∩ A = ∅ . Thus F ∩ A = ( F ∪ F ) ∩ A = ( F ∩ A ) ∪ ( F ∩ A ) = ∅ ,contradicting the assumption on F . Therefore, F is irreducible. C. Shen, X. Xi, X. Xu, and D. Zhao
Vol. 16:4 Saturated well-filtered spaces
In this section, we show that well-filteredness can be characterized by means of saturatedsets, instead of compact saturated sets.
Definition 3.1. A T space X is called saturated well-filtered , if for any { A i : i ∈ I } ⊆ flt S ( X ) and U ∈ O ( X ), (cid:92) i ∈ I A i ⊆ U ⇒ ∃ i ∈ I, A i ⊆ U. Definition 3.2. A T space X is called saturated ω -well-filtered , if for any { A n : n < ω } ⊆ flt S ( X ) and U ∈ O ( X ), (cid:92) n<ω A n ⊆ U ⇒ ∃ n < ω, A n ⊆ U. A countable family { A n : n < ω } ⊆ S ( X ) is called a descending (cid:28) -chain if A (cid:29) A (cid:29) A (cid:29) . . . (cid:29) A n (cid:29) A n +1 (cid:29) . . . . Analogous to Proposition 2.3, we can prove the following.
Proposition 3.3. A T space X is saturated ω -well-filtered if and only if for any countabledescending (cid:28) -chain { A n : n < ω } ⊆ S ( X ) and U ∈ O ( X ) , (cid:92) n<ω A n ⊆ U ⇒ ∃ n < ω, A n ⊆ U. Proposition 3.4.
Let X be a saturated well-filtered space. Then for any { A i : i ∈ I } ⊆ flt S ( X ) \ {∅} , (cid:84) i ∈ I A i is a nonempty compact saturated set.Proof. It is clear that (cid:84) i ∈ I A i is saturated. Now suppose that (cid:84) i ∈ I A i = ∅ . Since X issaturated well-filtered and ∅ is open, we have that A i ⊆ ∅ for some i ∈ I , which contradictsthat A i (cid:54) = ∅ . Thus (cid:84) i ∈ I A i (cid:54) = ∅ .Let { V j : j ∈ J } be an open cover of (cid:84) i ∈ I A i . As X is saturated well-filtered, thereexists i ∈ I such that A i ⊆ (cid:83) j ∈ J V j . Since { A i : i ∈ I } ⊆ S ( X ) is a (cid:28) -filtered family,there exists i ∈ I such that A i (cid:28) A i ⊆ (cid:83) j ∈ J V j . Then there exists J ⊆ J ( <ω ) such that A i ⊆ (cid:83) j ∈ J V j . It follows that (cid:84) i ∈ I A i ⊆ (cid:83) j ∈ J V j . Therefore, (cid:84) i ∈ I A i is compact.Using a similar proof to that of Proposition 3.4, we deduce the following. Proposition 3.5.
Let X be a saturated ω -well-filtered space. Then for any { A n : n <ω } ⊆ flt S ( X ) \ {∅} , (cid:84) n<ω A n is a nonempty compact saturated set. Theorem 3.6.
The saturated ω -well-filtered spaces are exactly the ω -well-filtered spaces.Proof. Note that every descending chain in Q ( X ) is a descending (cid:28) -chain in S ( X ). Henceevery saturated ω -well-filtered space is an ω -well-filtered space.Now let X be an ω -well-filtered space. Suppose that { A n : n < ω } ⊆ S ( X ) is adescending (cid:28) -chain, i.e., A (cid:29) A (cid:29) A (cid:29) . . . (cid:29) A n (cid:29) A n +1 (cid:29) . . . , and U ∈ O ( X ) such that (cid:84) n<ω A n ⊆ U . We need to prove that A n ⊆ U for some n < ω .Otherwise, A n (cid:42) U for all n < ω , that is, A n ∩ ( X \ U ) (cid:54) = ∅ . Then by Lemma 2.4, thereexists a minimal (irreducible) closed set F ⊆ X \ U such that F ∩ A n (cid:54) = ∅ for all n < ω .Choose one x n ∈ F ∩ A n for each n < ω , and let H := { x n : n < ω } . ol. 16:4 OPEN WELL-FILTERED SPACES 18:5 Claim: H is compact.Let { C i : i ∈ I } be a family of closed subsets of X such that for any J ∈ I ( <ω ) , H ∩ (cid:84) i ∈ J C i (cid:54) = ∅ . It needs to prove that H ∩ (cid:84) i ∈ I C i (cid:54) = ∅ . We complete the proof byconsidering two cases.(c1) C i ∩ H is infinite for all i ∈ I .In this case, for each n < ω , there exists k n ≥ n such that x k n ∈ C i . Since A k n ⊆ A n and x k n ∈ F ∩ A k n , we have that x k n ∈ C i ∩ F ∩ A k n ⊆ C i ∩ F ∩ A n (cid:54) = ∅ . Thus C i ∩ F is aclosed set that intersects all A n ( n < ω ). By the minimality of F , we have F = C i ∩ F , thatis, F ⊆ C i . By the arbitrariness of i ∈ I , it follows that F ⊆ (cid:84) i ∈ I C i . Note that H ⊆ F , so H ∩ (cid:84) i ∈ I C i = H (cid:54) = ∅ .(c2) C i ∩ H is finite for some i ∈ I .Let i ∈ I such that C i ∩ H is finite (hence compact). Note that the family { C i : i ∈ I } satisfies that for any J ∈ I ( <ω ) , H ∩ C i ∩ (cid:84) i ∈ J C i (cid:54) = ∅ . Since H ∩ C i is compact, weconclude H ∩ (cid:84) i ∈ I C i = ( H ∩ C i ) ∩ (cid:84) i ∈ I C i (cid:54) = ∅ .Now for each n < ω , let H n := { x k : k ≥ n } , which is compact by using a similar prooffor H . Then {↑ H k : k < ω } ⊆ flt Q ( X ) such that (cid:84) n<ω ↑ H n ⊆ (cid:84) n<ω A n ⊆ U . As X isan ω -well-filtered space, there exists n < ω such that ↑ H n ⊆ U , which contradicts that H n ⊆ F ⊆ X \ U . Theorem 3.7.
The saturated well-filtered spaces are exactly the well-filtered spaces.Proof.
Clearly, every saturated well-filtered space is a well-filtered space.Now assume that X is a well-filtered space. Let A ⊆ flt S ( X ) and U ∈ O ( X ) such that (cid:84) A ⊆ U .Define (cid:98) A = (cid:40) (cid:92) n<ω A n : ∀ n < ω, A n ∈ A and A n (cid:29) A n +1 (cid:41) . By Theorem 3.6 and the fact that every well-filtered space is ω -well-filtered, we deducethat X is saturated ω -well-filtered. Thus by Proposition 3.5, every member of (cid:98) A is nonemptycompact saturated.Claim: (cid:98) A is a filtered family.Let (cid:84) n<ω A n , (cid:84) n<ω B n ∈ (cid:98) A , that is, A (cid:29) A (cid:29) A (cid:29) · · · (cid:29) A n (cid:29) A n +1 (cid:29) . . . and B (cid:29) B (cid:29) B (cid:29) · · · (cid:29) B n (cid:29) B n +1 (cid:29) · · · . (i) There exists C ∈ A such that C (cid:28) A , B because A is (cid:28) -filtered.(ii) If we have defined { C , C , · · · C n } , then there exists A ∈ A such that A (cid:28) C n , A n +1 , B n +1 . Put C n +1 := A .By Induction, we obtain a family { C n : n < ω } ⊆ A such that C (cid:29) C (cid:29) C (cid:29) · · · C n (cid:29) C n +1 (cid:29) · · · , and that C n (cid:28) A n , B n (hence C n ⊆ A n , B n ) for all n < ω . It follows that (cid:84) n<ω C n ∈ (cid:98) A and it is a lower bound of (cid:8)(cid:84) n<ω A n , (cid:84) n<ω B n (cid:9) . Hence (cid:98) A is filtered.Since X is well-filtered and (cid:84) (cid:98) A = (cid:84) A ⊆ U , there is (cid:84) n<ω A n ∈ (cid:98) A , where A (cid:29) A (cid:29) A (cid:29) · · · (cid:29) A n (cid:29) A n +1 (cid:29) . . . , C. Shen, X. Xi, X. Xu, and D. Zhao
Vol. 16:4 such that (cid:84) n<ω A n ⊆ U . By Theorem 3.6, X is saturated ω -well-filtered. Then there exists n < ω such that A n ⊆ U (note that A n ∈ A ). Therefore, X is well-filtered.4. Open well-filtered spaces
In this section, we define another class of T spaces, called open well-filtered spaces. Itturns out that every well-filtered space is open well-filtered, and every core-compact openwell-filtered space is sober. As an immediate consequence, we have that every core-compactwell-filtered space is sober. Definition 4.1. A T space is called open well-filtered , if for any { U i : i ∈ I } ⊆ flt O ( X )and U ∈ O ( X ), (cid:92) i ∈ I U i ⊆ U ⇒ ∃ i ∈ I, U i ⊆ U. Note that every open set is saturated. By Theorem 3.7, we obtain the following result.
Remark 4.2.
Every well-filtered space is open well-filtered.
Definition 4.3. A T space X is called open ω -well-filtered , if for any { U n : n < ω } ⊆ flt O ( X ) and U ∈ O ( X ), (cid:92) n<ω U n ⊆ U ⇒ ∃ n < ω, U n ⊆ U. Using a similar proof to that of Proposition 2.3, we deduce the following result.
Proposition 4.4. A T space X is open ω -well-filtered if and only if for any countabledescending (cid:28) -chain { U n : n < ω } ⊆ O ( X ) and U ∈ O ( X ) , (cid:92) n<ω U n ⊆ U ⇒ ∃ n < ω, U n ⊆ U. Analogous to Proposition 3.4, we have the following two results.
Proposition 4.5.
Let X be an open well-filtered space. Then for any { U i : i ∈ I } ⊆ flt O ( X ) , (cid:84) i ∈ I U i is a nonempty compact saturated set. Proposition 4.6.
Let X be an open ω -well-filtered space. Then for any { U n : n < ω } ⊆ flt O ( X ) , (cid:84) n<ω U n is a nonempty compact saturated set. Theorem 4.7.
Every core-compact open well-filtered space is sober.Proof.
Assume X is a core-compact open well-filtered space. Let A be an irreducible closedsubset of X . Define F A := { U ∈ O ( X ) : U ∩ A (cid:54) = ∅} . Claim: F A is a (cid:28) -filtered family.Let U , U ∈ F A . Then U ∩ A (cid:54) = ∅ (cid:54) = U ∩ A . As A is irreducible, U ∩ U ∩ A (cid:54) = ∅ , sothere is an x ∈ A ∩ U ∩ U . Since X is core-compact, there exists U ∈ O ( X ) such that x ∈ U (cid:28) U ∩ U . Note that x ∈ U ∩ A (cid:54) = ∅ , so U ∈ F A . Hence, F A is a (cid:28) -filtered family.Since X is open well-filtered, A ∩ (cid:84) i ∈ I F A (cid:54) = ∅ . Let x ∈ A ∩ (cid:84) F A . We showthat A = cl( { x } ). Otherwise, A \ cl( { x } ) = A ∩ ( X \ cl( { x } )) (cid:54) = ∅ , implying that X \ cl( { x } ) ∈ F A . It follows that x ∈ (cid:84) F A ⊆ X \ cl( { x } ), a contradiction. Thus A = cl( { x } ). Since X is a T space, { x } is unique. So X is sober. ol. 16:4 OPEN WELL-FILTERED SPACES 18:7 As a consequence of Remark 4.2 and Theorem 4.7, we obtain the following result.
Corollary 4.8.
Every core-compact well-filtered space is sober.
Theorem 4.9.
Every core-compact open ω -well-filtered space is locally compact.Proof. Assume that X is a core-compact open ω -well-filtered space. Let x ∈ X and U ∈ O ( X )such that x ∈ U . Since X is core-compact, there exists an open set W (cid:28) U such that x ∈ W and a sequence of open sets { U n : n < ω } such that U = U (cid:29) U (cid:29) U (cid:29) U (cid:29) . . . (cid:29) W. Let Q = (cid:84) n<ω U n . Since X is open ω -well-filtered and by Proposition 4.5, Q ∈ Q ( X )satisfies that x ∈ W ⊆ Q ⊆ U . Thus X is locally compact. Remark 4.10.
In [GL19], J. Goubault-Larrecq gives a slighly different proof for the abovetheorem.As a corollary of Theorem 4.9, we deduce the following result.
Corollary 4.11.
A well-filtered space is core-compact if and only if it is locally compact.
The following result is a small variant of Kou’s result that every well-filtered space is a d -space (see Proposition 2.4 in [Kou01]). Proposition 4.12 [Kou01] . Let X be an ω -well-filtered space. If D is a directed (under thespecialization order) subset of X with the cardinality | D | ≤ ω , then (cid:87) D exists. Let P be a poset. A subset U of P is Scott open if (i) U = ↑ U and (ii) for any directedsubset D of P for which (cid:87) D exists, (cid:87) D ∈ U implies D ∩ U (cid:54) = ∅ . All Scott open subsetsof P form a topology, called the Scott topology on P and denoted by σ ( P ). The spaceΣ P = ( P, σ ( P )) is called the Scott space of P . Example 4.13.
Let J = N × ( N ∪ { ω } ) be the Johnstone’s dcpo and N = { , , . . . } withthe usual order. Let P = J ∪ N . For any x, y ∈ P , define x ≤ y (refer to Figure 1) if one ofthe following conditions holds:(i) x, y ∈ N and x ≤ y in N ;(ii) x, y ∈ J and x ≤ y in J ;(iii) x ∈ N , y ∈ J and y = ( x, ω ).Next, we show that Σ P is an open well-filtered space. The following conclusions on P will be used later.(c1) For any directed subset D of P , if (cid:87) D exists, then (cid:87) D ∈ D or D ⊆ J .In fact, assume D (cid:42) J , that is, D ∩ N (cid:54) = ∅ . We prove (cid:87) D ∈ D by considering thefollowing two cases:Case 1: D ⊆ N . In this case, it is trivial that (cid:87) D ∈ D .Case 2: D (cid:42) N , that is, D ∩ N (cid:54) = ∅ and D ∩ J (cid:54) = ∅ . Let k ∈ D ∩ N and ( m, n ) ∈ D ∩ J .Since D is directed, there exists d ∈ D such that k, ( m, n ) ≤ d . It forces that d = ( n , ω )for some n ∈ N , which is a maximal point of P , so d = (cid:87) D ∈ D (note that themaximal point of a directed set is exactly the least upper bound of the set).(c2) For any U ∈ σ ( P ) \ {∅} , there exists a minimal n U ∈ N such that ( n, ω ) ∈ U for all n ≥ n U , that is, n U = min { k ∈ N : ∀ n ≥ k, ( n, w ) ∈ U } exists in N (refer to Figure 1). C. Shen, X. Xi, X. Xu, and D. Zhao
Vol. 16:4
Figure 1: The poset P Assume, on the contrary, that n U does not exist in N . Then there exist infinitelycountable numbers n , n , n , . . . in N such that ( n k , ω ) / ∈ U for all k ∈ N . Since U is a nonempty upper set, there exists m ∈ N such that ( m, ω ) ∈ U . Note that { ( m, n k ) : k ∈ N } is a directed set such that (cid:87) { ( m, n k ) : k ∈ N } = ( m, ω ) ∈ U . Thenthere exists k ∈ N such that ( m, n k ) ∈ U . Since ( n k , ω ) ≥ ( m, n k ) ∈ U and U is anupper set, it holds that ( n k , ω ) ∈ U , a contradiction.(c3) For any U ∈ σ ( P ) \ {∅} and ∀ ( n, ω ) ∈ U , there exists a minimal φ U ( n ) ∈ N such that( n, φ U ( n )) ∈ U , i.e., φ U ( n ) = min { m ∈ N : ( n, m ) ∈ U } exists.Suppose ( n, ω ) ∈ U . Then { ( n, m ) : m ∈ N } is a directed subset of P such that (cid:87) { ( n, m ) : m ∈ N } = ( n, ω ) ∈ U . Since U is Scott open, there exists m ∈ N such that ( n, m ) ∈ U . Thus m ∈ { m ∈ N : ( n, m ) ∈ U } (cid:54) = ∅ , which means that ϕ U ( n ) = min { m ∈ N : ( n, m ) ∈ U } exists.(c4) For any U, V ∈ σ ( P ), U (cid:28) V if and only if U = ∅ .If U = ∅ , then trivially U = ∅ (cid:28) V . Now assume U (cid:28) V and U (cid:54) = ∅ . By using(c2) and (c3), for each n ≥ n U , define U n = P \ (cid:83) k ≥ n ↓ ( k, φ U ( k )). It is trivial that (cid:83) k ≥ n ↓ ( k, φ U ( k )) is a Scott closed subset of P , which means that U n ∈ σ ( P ). Since (cid:83) n ≥ n U P \ (cid:83) k ≥ n ↓ ( k, φ U ( k )) = P \ (cid:84) n ≥ n U (cid:83) k ≥ n ↓ ( k, φ U ( k )) = P \ ∅ = P . Thus thefamily { U n : n ≥ n U } is a directed open cover of P , hence a directed open cover of V . By assumption that U (cid:28) V , there exists n ≥ n U such that U ⊆ U n . By thedefinition of n U , it follows that ( n , ω ) ∈ U . Thus by the definition of φ U in (c3), wehave that ( n , φ U ( n )) ∈ U , but ( n , φ U ( n )) / ∈ P \ ↓ ( n , φ U ( n )) ⊇ U n , which implies( n , φ U ( n )) / ∈ U n . It follows that ( n , φ U ( n )) ∈ U \ U n , contradicting that U ⊆ U n .By (c4), we conclude that every (cid:28) -filtered family F of Scott open subsets of P contains ∅ , the minimal element in F . This implies that Σ P is an open well-filtered space (hence anopen ω -well-filtered space). Since (cid:87) N does not exist in P and by Proposition 4.12, Σ P isnot an ω -well-filtered space. ol. 16:4 OPEN WELL-FILTERED SPACES 18:9 Recall that a T space X is called a d -space if X is a dcpo in its specialization orderand each open subset of X is Scott open in the specialization order. It is well-known thatevery well-filtered space is a d -space [GL13]. From Example 4.13, we have the followingconclusions.(1) An open well-filterd space need not be a d -space.(2) An open well-filtered space need not be an ω -well-filtered space.A summary on the relations among kinds of well-filtered spaces is shown in Figure 2. sober (cid:15) (cid:15) well-filtered (cid:111) (cid:111) (cid:47) (cid:47) (cid:15) (cid:15) saturatedwell-filtered (cid:47) (cid:47) (cid:15) (cid:15) openwell-filtered (cid:15) (cid:15) core-compact (cid:107) (cid:107) saturated ω -well-filtered (cid:111) (cid:111) (cid:47) (cid:47) ω -well-filtered (cid:47) (cid:47) open ω -well-filteredcore-compact (cid:47) (cid:47) locallycompact Figure 2: The relations among various types of well-filtered spaces5.
Acknowledgment
We would like to thank the reviewers for giving us valuable comments and suggestions forimproving the manuscript.
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