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Selectors for sequences of subsets of hyperspaces
Journal Pre-proof Selectors for sequences of subsets of hyperspaces
Alexander V. Osipov
PII:
S0166-8641(19)30413-4
DOI:
https://doi.org/10.1016/j.topol.2019.107007
Reference:
TOPOL 107007
To appear in:
Topology and its Applications
Received date:
2 November 2018
Revised date:
21 March 2019
Accepted date:
26 March 2019
Please cite this article as: A.V. Osipov, Selectors for sequences of subsets of hyperspaces, Topol. Appl. (2019), 107007, doi: https://doi.org/10.1016/j.topol.2019.107007.
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Selectors for sequences of subsets of hyperspaces Alexander V. Osipov Krasovskii Institute of Mathematics and Mechanics, Ural Federal University, Ural State University of Economics, 620219, Yekaterinburg, Russia
Abstract For a Hausdorff space X we denote by 2X the family of all closed subsets of X. In this paper we continue to research relationships between closure-type properties of hyperspaces over a space X and covering properties of X. We investigate selectors for sequence of subsets of the space 2X with the Z+ topology and the upper Fell topology (F+ -topology). Also we consider the selection properties of the bitopological space (2X , F+ , Z+ ). Keywords: hyperspace, upper Fell topology, selection principles, bitopological space, Z+ -topology, perfect space, k-perfect space 2010 MSC: 54B20, 54D20, 54E55 1. Introduction Given a Hausdorff space X we denote by 2X the family of all closed subsets of X. If A is a subset of X and A a family of subsets of X, then Ac = X \ A and Ac = {Ac : A ∈ A}, A− = {F ∈ 2X : F ∩ A = ∅}, A+ = {F ∈ 2X : F ⊂ A}. Let Δ be a subset of 2X closed for finite unions and containing all singletons. We consider the next important cases: • Δ is the collection CL(X) = 2X \ {∅}; • Δ is the family K(X) of all non-empty compact subsets of X; • Δ is the family F(X) of all non-empty finite subsets of X. For Δ ⊂ 2X , the upper Δ-topology, denoted by Δ+ , is the topology whose base is the collection {(Dc )+ : D ∈ Δ} ∪ {2X }. Email address: [email protected] (Alexander V. Osipov) Preprint submitted to ...
December 10, 2019
When Δ = CL(X) we have the well-known upper Vietoris topology V+ , when Δ = K(X) we have the upper Fell topology (known also as the cocompact topology) F+ , and when Δ = F(X) we have the Z+ -topology. Many topological properties are defined or characterized in terms of the following classical selection principles ([4, 17, 19]). Let A and B be sets consisting of families of subsets of an infinite set X. Then: S1 (A, B) is the selection hypothesis: for each sequence (An : n ∈ N) of elements of A there is a sequence (bn : n ∈ N) such that for each n, bn ∈ An , and {bn : n ∈ N} is an element of B. Sf in (A, B) is the selection hypothesis: for each sequence (An : n ∈ N) of elements of A there isa sequence (Bn : n ∈ N) of finite sets such that for each n, Bn ⊆ An , and n∈N Bn ∈ B. In this paper, by a cover we mean a nontrivial one, that is, U is a cover of X if X = U and X ∈ / U. An open cover of a space is large if each element of the space belongs to infinitely many elements of the cover. An open cover U of a space X is called: • an ω-cover (a k-cover) if each finite (compact) subset C of X is contained in an element of U; • a γ-cover (a γk -cover) if U is infinite and for each finite (compact) subset C of X the set {U ∈ U : C U } is finite. Because of these definitions all spaces are assumed to be Hausdorff noncompact, unless otherwise stated. Let us mention that any ω-cover (k-cover) is infinite and large, and that any infinite subfamily of a γ-cover (γk -cover) is also a γ-cover (γk -cover). For a topological space X we denote: • O — the family of all open covers of X; • Γ — the family of all open γ-covers of X; • Γk — the family of all open γk -covers of X; • Ω — the family of all open ω-covers of X; • K — the family of all open k-covers of X. Different Δ-covers (k-covers, ω-covers, kF -covers, cF -covers,...) exposed many dualities in hyperspace topologies such as co-compact topology F+ , cofinite topology Z+ , Pixley-Roy topology, Fell topology and Vietoris topology. They also play important roles in selection principles ([1,4-9,12-16]). In this paper we continue to research relationships between closure-type properties of hyperspaces over a space X and covering properties of X. We 2
investigate selectors for sequence of subsets of the space 2X with the Z+ topology and the upper Fell topology (F+ -topology). Also we consider the selection properties of the bitopological space (2X , F+ , Z+ ). 2. Main definitions and notation The following lammas will be often used throughout the paper, sometimes without explicit reference. Lemma 2.1. (Lemma 1 in [1]) Let Y be an open subset of a space X and U an open cover of Y . Then the following holds: (1) U is a k-cover of Y ⇔ Y c ∈ ClF + (U c ); (2) U is an ω-cover of Y ⇔ Y c ∈ ClZ + (U c ). Lemma 2.2. (Lemma 2 in [1]) Let X be a topological space, Y an open subsets of X and U = {Un : n ∈ N} an open cover of Y . Then (1) U is a γk -cover of Y ⇔ the sequence (Unc : n ∈ N) converges to Y c in (2X , F+ ); (2) U is an γ-cover of Y ⇔ the sequence (Unc : n ∈ N) converges to Y c in X (2 , Z+ ). Lemma 2.3. (Lemma 3 in [1]) Given a space X and an open cover U of X the following holds: (1) U is a k-cover of X ⇔ U c is a dense subset of (2X , F+ ); (2) U is an ω-cover of X ⇔ U c is a dense subset of (2X , Z+ ). Let X be a topological space, and x ∈ X. A subset A of X converges to x, x = lim A, if A is infinite, x ∈ / A, and for each neighborhood U of x, A \ U is finite. Consider the following collection: • Ωx = {A ⊆ X : x ∈ A \ A}; • Γx = {A ⊆ X : x = lim A}. Note that if A ∈ Γx , then there exists {an } ⊂ A converging to x. So, simply Γx may be the set of non-trivial convergent sequences to x. Definition 2.4. Let X be a space and let U = {Uα : α ∈ Λ} be an open cover of X. Then U c = {Uαc : α ∈ Λ} converges to {∅} in (2X , τ ) where τ is a topology on 2X , if for every F ∈ 2X the U c converges to F , i.e. for each neighborhood W of F in the space (2X , τ ), |{α : Uαc W, α ∈ Λ}| < ℵ0 . Lemma 2.5. Let X be a space and let U = {Uα : α ∈ Λ} be an open cover of X. Then the following are equivalent: 3
1. U is an γ-cover of X; 2. U c converges to {∅} in (2X , Z+ ). For a topological space (2X , τ ) we denote: • DΩ — the family of dense subsets of (2X , τ ); • DΓ — the family of converging to {∅} subsets of (2X , τ ). Since every γ-cover contains a countably γ-cover, then each converging to {∅} subset of (2X , Z+ ) contains a countable converging to {∅} subset of (2X , Z+ ). 3. Hyperspace (2X , Z+ ) Theorem 3.1. Assume that Φ, Ψ ∈ {Γ, Ω}, ∈ {1, f in}. Then for a space X the following statements are equivalent: 1. X satisfies S (Φ, Ψ); 2. (2X , Z+ ) satisfies S (DΦ , DΨ ). Proof. We prove the theorem for = f in, the other proofs being similar. (1) ⇒ (2). Let (Di : i ∈ N) be a sequence of dense subsets of (2X , Z+ ) such that Di ∈ DΦ for each i ∈ N. Then (Dic : i ∈ N) is a sequence of open covers of X such that Dic ∈ Φ for each i ∈ N. Since X satisfies Sf in (Φ, Ψ), thereis a sequence (Ai : i ∈ N) offinite sets such that for each i, Ai ⊆ Dic , and i∈N Ai ∈ Ψ. It follows that i∈N Aci ∈ DΨ . (2) ⇒ (1). Let (Un : n ∈ N) be a sequence of open covers of X such that Un ∈ Φ. For each n, An := Unc is a dense subset of (2X , Z+ ) such that An ∈ DΦ . Applying that (2X , Z+ ) satisfies Sf in (DΦ , DΨ ), there is a sequence (An : n∈ N) of finite sets such that for each n, An ⊆ An , and n∈N An ∈ DΨ . Then n∈N Un is an open cover of X where Un = Acn for each n ∈ N and n∈N Un ∈ Ψ. Corollary 3.2. (Theorem 5 in [8]) For a space X the following are equivalent: 1. X satisfies S1 (Ω, Ω); 2. (2X , Z+ ) satisfies S1 (DΩ , DΩ ). Corollary 3.3. (Theorem 13 in [8]) For a space X the following are equivalent: 4
1. X satisfies Sf in (Ω, Ω); 2. (2X , Z+ ) satisfies Sf in (DΩ , DΩ ). Theorem 3.4. Assume that Φ, Ψ ∈ {Γ, Ω, }, ∈ {1, f in}. Then for a space X the following statements are equivalent: 1. Each open set Y ⊂ X has the property S (Φ, Ψ); 2. For each E ∈ 2X , (2X , Z+ ) satisfies S (ΦE , ΨE ). Proof. We prove the theorem for = 1, the other proofs being similar. (1) ⇒ (2). Let E ∈ 2X and let (An : n ∈ N) be a sequence such that An ∈ ΦE for each n ∈ N. Then (Acn : n ∈ N) is a sequence of open covers of E c such that Acn ∈ Φ for each n ∈ N. Since E c has the property S (Φ, Ψ), there is a sequence (Acn : n ∈ N) such that Acn ∈ Acn for each n ∈ N and {Acn : n ∈ N} is open cover of E c such that {Acn : n ∈ N} ∈ Ψ. It follows that {An : n ∈ N} ∈ ΨE . (2) ⇒ (1). Let Y be an open subset of X and let (Fn : n ∈ N) be a sequence of open covers of Y such that Fn ∈ ΦY (where ΦY is the Φ family of covers of Y ). Let E = X \Y . Put An = Fnc for each n ∈ N. Then An ⊂ 2X and An ∈ ΦE for each n ∈ N. Since, by (2), (2X , Z+ ) satisfies S1 (ΦE , ΨE ), there is a sequence (An : n ∈ N) such that An ∈ An for each n ∈ N and {An : n ∈ N} ∈ ΨE . It follows that {Fn : Fn = Acn , n ∈ N} ∈ Ψ. Corollary 3.5. (Theorem 3 in [5]) For a space X the following statements are equivalent: 1. Each open set Y ⊂ X has the property S1 (Ω, Γ); 2. (2X , Z+ ) is Fr´ echet-Urysohn; X + 3. (2 , Z ) is strongly Fr´ echet-Urysohn. Corollary 3.6. (Theorem 1 in [8]) For a space X the following are equivalent: 1. Each open set Y ⊂ X satisfies S1 (Ω, Ω); 2. (2X , Z+ ) has countable strong fan tightness (For each E ∈ 2X , (2X , Z+ ) satisfies S1 (ΩE , ΩE )). Corollary 3.7. (Theorem 9 in [8]) For a space X the following are equivalent: 1. Each open set Y ⊂ X satisfies Sf in (Ω, Ω); 5
2. (2X , Z+ ) has countable fan tightness (For each E ∈ 2X , (2X , Z+ ) satisfies Sf in (ΩE , ΩE )). Recall that a space is perfect if every open subset is an Fσ -subset [10]. Clearly every semi-stratifiable space is perfect. Note that all properties in the Scheepers Diagram ([3, 18]) are hereditary for Fσ -subsets (Corollary 2.4 in [11]). Proposition 3.8. Assume that Φ, Ψ ∈ {Γ, Ω, }, ∈ {1, f in}. Then for a perfect topological space X the following statements are equivalent: 1. X satisfies S (Φ, Ψ); 2. Each open set Y ⊂ X has the property S (Φ, Ψ). Theorem 3.9. Assume that Φ, Ψ ∈ {Γ, Ω}, ∈ {1, f in}. Then for a perfect space X the following statements are equivalent: 1. X satisfies S (Φ, Ψ); 2. (2X , Z+ ) satisfies S (DΦ , DΨ ); 3. For each E ∈ 2X , (2X , Z+ ) satisfies S (ΦE , ΨE ). Clearly that every perfectly normal space is perfect. Corollary 3.10. Assume that Φ, Ψ ∈ {Γ, Ω}, ∈ {1, f in}. Then for a perfectly normal space X the following statements are equivalent: 1. X satisfies S (Φ, Ψ); 2. (2X , Z+ ) satisfies S (DΦ , DΨ ); 3. For each E ∈ 2X , (2X , Z+ ) satisfies S (ΦE , ΨE ). 4. Hyperspace (2X , F+ ) Note that (2X , F+ ) satisfies the selection principle Sf in (O, O) for any space X, because (2X , F+ ) is always compact (see [2]). Sf in (O, O) property is called the Menger property (see [3, 18]). Lemma 4.1. Let X be a space and let U = {Uα : α ∈ Λ} be an open cover of X. Then the following are equivalent: 1. U is an γk -cover of X; 2. U c converges to {∅} in (2X , F+ ). 6
Theorem 4.2. Assume that Φ, Ψ ∈ {Γk , K}, ∈ {1, f in}. Then for a space X the following statements are equivalent: 1. X satisfies S (Φ, Ψ); 2. (2X , F+ ) satisfies S (DΦ , DΨ ). Proof. The proof is similar to the proof of Theorem 3.1. Corollary 4.3. (Theorem 4 in [8]) For a space X the following are equivalent: 1. X satisfies S1 (K, K); 2. (2X , F+ ) satisfies S1 (DΩ , DΩ ). Corollary 4.4. (Theorem 12 in [8]) For a space X the following are equivalent: 1. X satisfies Sf in (K, K); 2. (2X , F+ ) satisfies Sf in (DΩ , DΩ ). Theorem 4.5. Assume that Φ, Ψ ∈ {Γk , K}, ∈ {1, f in}. Then for a space X the following statements are equivalent: 1. Each open set Y ⊂ X has the property S (Φ, Ψ); 2. For each E ∈ 2X , (2X , F+ ) satisfies S (ΦE , ΨE ). Proof. The proof is similar to the proof of Theorem 3.4. Corollary 4.6. (Theorem 23 in [1]) For a space X the following are equivalent: 1. For each E ∈ 2X , (2X , F+ ) satisfies S1 (ΓE , ΩE ); 2. Each open set Y ⊂ X has the property S1 (Γk , K). Corollary 4.7. (Theorem 32 in [1]) For a space X the following are equivalent: 1. For each E ∈ 2X , (2X , F+ ) satisfies S1 (ΓE , ΓE ); 2. Each open set Y ⊂ X has the property S1 (Γk , Γk ).
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Definition 4.8. A subset A of a space X is called an k-Fσ -set if A can be ∞ Fi where Fi is a closed set in X for each i ∈ N and represented as A = i=1
for any compact set B ⊆ A there exists i ∈ N such that B ⊆ Fi . Definition 4.9. A space X is called k-perfect if every open subset is an k-Fσ -subset of X. Clearly that every k-perfect space is perfect. Proposition 4.10. Every perfectly normal space is k-perfect. Proof. Let U be an open subset of a perfectly normal space X. Then U ∞ may be represented as U = Fi where Fi is closed set of X for each i ∈ N i=1
and Fi ⊂ Fi+1 . Since X is a perfectly normal space, there exists a sequence (Ui : i ∈ N) of open sets of X such that Fi ⊆ Ui ⊆ Ui ⊆ U and Ui ⊆ Ui+1 for ∞ Ui and for any compact set B ⊆ U there each i ∈ N. It follow that U = exists i ∈ N such that B ⊆ Ui .
i=1
Note that, by Proposition 4.10, each cozero set is an k-Fσ -set. Question. Is there a k-perfect which is not (perfectly) normal space? Denote by S the Sorgenfrey line. Proposition 4.11. The space S2 is perfect, but not k-perfect. Proof. By Lemma 2.3 in [10], S2 is perfect. Consider the open set U = {(x, y) ∈ S2 : −x < y} {(x, −x) : x ∈ P} in the space S2 . ∞ Fi where Fi is closed Assume that U is a k-Fσ -set of S2 . Then U = i=1
set in S2 for each i ∈ N and for any compact set B ⊆ U there exists i ∈ N such that B ⊆ Fi . For each point1p = (y, 1−y) ∈ {(x, −x) : x ∈ P} we fix the compact set Zp = {p} {(y + n , −y + n ) : n ∈ N}. Note that Zp ⊂ U ∞ for each p ∈ {(x, −x) : x ∈ P}. Since U = Fi there exists k ∈ N such i=1
that |{p : Zp ⊂ Fk }| > ℵ0 . Since the set {p : Zp ⊂ Fk } is uncountable 8
subset of {(x, −x) : x ∈ P} there is a accumulation point z in subspace {(x, −x) : x ∈ R} of the space R2 . Clearly that for any neighborhood O(z) of z in the space S2 we have that O(z) Zp = ∅. It follows that Zp ⊂Fk O(z) Fk = ∅. Hence Fk is not closed set in S2 , a contradiction. Theorem 4.12. Assume that Φ, Ψ ∈ {Γk , K}, ∈ {1, f in}, X has the property S (Φ, Ψ) and A is an k-Fσ -set. Then A has the property S (Φ, Ψ). Proof. We prove the theorem for = f in, the other proofs being similar. Assume that X has the property Sf in (Φ, Ψ) and A is an k-Fσ -set. Consider a sequence (Ui : i ∈ N) of covers A such that Ui ∈ ΦA (where ΦA is the ∞ Fi where Fi is a closed Φ family of covers of A) for each i ∈ N. Let A = i=1
set in X for each i ∈ N and for any compact set B ⊆ A there exists i ∈ N such that B ⊆ Fi . Consider Vi = {(X \ Fi ) U : U ∈ Ui } for each i ∈ N. We claim that Vi ∈ Φ for each i ∈ N. Let S be a compact subset of X. Then U ∈ Ui such that hence there is S Fi is a compact subset of A and S Fi ⊂ U . It follows that S ⊂ (X \ Fi ) U for (X \ Fi ) U ∈ Vi . Since X has the property Sf in (Φ, Ψ), there is a sequence (Bi : i ∈ N) of finite sets such thatfor eachi, Bi ⊂ V , and i i∈N Bi ∈ Ψ. We claim that i∈N {Bi Fi : Bi Fi ⊂ Ui , i ∈ N} ∈ ΨA . Let B be a compact subset of A then there exists i ∈ N such that B ⊆ Fi . Since i∈N Bi is a large cover of X there is k ∈N and Vk ∈ Bk ⊂ Vk such that k > i and B ⊂ Vk . But Vk = (X \ Fk ) Uk for Uk ∈ Uk . Since k > i , B ⊂ Uk . It follows that A has the property Sf in (Φ, Ψ).
Proposition 4.13. Assume that Φ, Ψ ∈ {Γk , K}, ∈ {1, f in}. Then for a k-perfect space X the following statements are equivalent: 1. X satisfies S (Φ, Ψ); 2. Each open set Y ⊂ X has the property S (Φ, Ψ). By Proposition 4.10, we have the following result. Proposition 4.14. Assume that Φ, Ψ ∈ {Γk , K}, ∈ {1, f in}. Then for a perfectly normal space X the following statements are equivalent: 1. X satisfies S (Φ, Ψ); 9
2. Each open set Y ⊂ X has the property S (Φ, Ψ). Theorem 4.15. Assume that Φ, Ψ ∈ {Γk , K}, ∈ {1, f in}. Then for a k-perfect space X the following statements are equivalent: 1. X satisfies S (Φ, Ψ); 2. (2X , F+ ) satisfies S (DΦ , DΨ ); 3. For each E ∈ 2X , (2X , F+ ) satisfies S (ΦE , ΨE ). By Proposition 4.10, we have the following result. Corollary 4.16. Assume that Φ, Ψ ∈ {Γk , K}, ∈ {1, f in}. Then for a perfectly normal space X the following statements are equivalent: 1. X satisfies S (Φ, Ψ); 2. (2X , F+ ) satisfies S (DΦ , DΨ ); 3. For each E ∈ 2X , (2X , F+ ) satisfies S (ΦE , ΨE ). 5. Bitopological space (2X , F+ , Z+ ) Theorem 5.1. Assume that Φ ∈ {Γk , K}, Ψ ∈ {Γ, Ω}, ∈ {1, f in}. Then for a space X the following statements are equivalent: 1. X satisfies S (Φ, Ψ); + Z+ ). 2. (2X , F+ , Z+ ) satisfies S (DΦF , DΨ Proof. The proof is similar to the proof of Theorem 3.1. Corollary 5.2. (Theorem 6 in [8]) For a space X the following are equivalent: 1. X satisfies S1 (K, Ω); + + 2. (2X , F+ , Z+ ) satisfies S1 (DΩF , DΩZ ). Corollary 5.3. (Theorem 14 in [8]) For a space X the following are equivalent: 1. X satisfies Sf in (K, Ω); + + 2. (2X , F+ , Z+ ) satisfies Sf in (DΩF , DΩZ ). Theorem 5.4. Assume that Φ ∈ {Γk , K}, Ψ ∈ {Γ, Ω}, ∈ {1, f in}. Then for a space X the following statements are equivalent: 10
1. Each open set Y ⊂ X has the property S (Φ, Ψ); + Z+ 2. For each E ∈ 2X , (2X , F+ , Z+ ) satisfies S (ΦF E , ΨE ). Proof. The proof is similar to the proof of Theorem 3.4. Corollary 5.5. (Theorem 31 in [1]) For a space X the following are equivalent: 1. Each open set Y ⊂ X has the property S1 (Γk , Ω); + Z+ 2. For each E ∈ 2X , (2X , F+ , Z+ ) satisfies S1 (ΓF E , ΩE ). Corollary 5.6. (Theorem 33 in [1]) For a space X the following are equivalent: 1. Each open set Y ⊂ X has the property S1 (Γk , Γ); + Z+ 2. For each E ∈ 2X , (2X , F+ , Z+ ) satisfies S1 (ΓF E , ΓE ). Theorem 5.7. Assume that Φ ∈ {Γk , K}, Ψ ∈ {Γ, Ω}, ∈ {1, f in}, X has the property S (Φ, Ψ) and A is an k-Fσ -set. Then A has the property S (Φ, Ψ). Proof. The proof is similar to the proof of Theorem 4.12. Proposition 5.8. Assume that Φ ∈ {Γk , K}, Ψ ∈ {Γ, Ω}, ∈ {1, f in}. Then for a k-perfect space X the following statements are equivalent: 1. X satisfies S (Φ, Ψ); 2. Each open set Y ⊂ X has the property S (Φ, Ψ). Theorem 5.9. Assume that Φ ∈ {Γk , K}, Ψ ∈ {Γ, Ω}, ∈ {1, f in}. Then for a k-perfect space X the following statements are equivalent: 1. X satisfies S (Φ, Ψ); 2. (2X , F+ , Z+ ) satisfies S (DΦ , DΨ ); 3. For each E ∈ 2X , (2X , F+ , Z+ ) satisfies S (ΦE , ΨE ).
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[13] A.V. Osipov, Classification of selectors for sequences of dense sets of Cp (X), Topology and its Applications, 242, (2018), 20-32. [14] A.V. Osipov, The functional characteristics of the Rothberger and Menger properties, Topology and its Applications, 243, (2018), 146–152. ¨ ca˘g, Variations of selective separability and tightness [15] A.V. Osipov, S. Oz¸ in function spaces with set-open topologies, Topology and its Applications, 217, (2017), p.38–50. [16] M. Sakai, Selective separability of Pixley-Roy hyperspaces, Topology and its Applications, 159, (2012), 1591–1598. [17] M. Sakai, M. Scheepers, The combinatorics of open covers, Recent Progress in General Topology III, (2013), Chapter, p. 751–799. [18] M. Scheepers, Combinatorics of open covers (I): Ramsey Theory, Topology and its Applications, 69, (1996), p. 31–62. [19] B. Tsaban, Some New Directions in Infinite-combinatorial Topology, in: Bagaria J., Todorcevic S. (eds) Set Theory. Trends in Mathematics. Birkh¨ auser Basel.(2006).
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Alexander V. Osipov
PII:
S0166-8641(19)30413-4
DOI:
https://doi.org/10.1016/j.topol.2019.107007
Reference:
TOPOL 107007
To appear in:
Topology and its Applications
Received date:
2 November 2018
Revised date:
21 March 2019
Accepted date:
26 March 2019
Please cite this article as: A.V. Osipov, Selectors for sequences of subsets of hyperspaces, Topol. Appl. (2019), 107007, doi: https://doi.org/10.1016/j.topol.2019.107007.
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Selectors for sequences of subsets of hyperspaces Alexander V. Osipov Krasovskii Institute of Mathematics and Mechanics, Ural Federal University, Ural State University of Economics, 620219, Yekaterinburg, Russia
Abstract For a Hausdorff space X we denote by 2X the family of all closed subsets of X. In this paper we continue to research relationships between closure-type properties of hyperspaces over a space X and covering properties of X. We investigate selectors for sequence of subsets of the space 2X with the Z+ topology and the upper Fell topology (F+ -topology). Also we consider the selection properties of the bitopological space (2X , F+ , Z+ ). Keywords: hyperspace, upper Fell topology, selection principles, bitopological space, Z+ -topology, perfect space, k-perfect space 2010 MSC: 54B20, 54D20, 54E55 1. Introduction Given a Hausdorff space X we denote by 2X the family of all closed subsets of X. If A is a subset of X and A a family of subsets of X, then Ac = X \ A and Ac = {Ac : A ∈ A}, A− = {F ∈ 2X : F ∩ A = ∅}, A+ = {F ∈ 2X : F ⊂ A}. Let Δ be a subset of 2X closed for finite unions and containing all singletons. We consider the next important cases: • Δ is the collection CL(X) = 2X \ {∅}; • Δ is the family K(X) of all non-empty compact subsets of X; • Δ is the family F(X) of all non-empty finite subsets of X. For Δ ⊂ 2X , the upper Δ-topology, denoted by Δ+ , is the topology whose base is the collection {(Dc )+ : D ∈ Δ} ∪ {2X }. Email address: [email protected] (Alexander V. Osipov) Preprint submitted to ...
December 10, 2019
When Δ = CL(X) we have the well-known upper Vietoris topology V+ , when Δ = K(X) we have the upper Fell topology (known also as the cocompact topology) F+ , and when Δ = F(X) we have the Z+ -topology. Many topological properties are defined or characterized in terms of the following classical selection principles ([4, 17, 19]). Let A and B be sets consisting of families of subsets of an infinite set X. Then: S1 (A, B) is the selection hypothesis: for each sequence (An : n ∈ N) of elements of A there is a sequence (bn : n ∈ N) such that for each n, bn ∈ An , and {bn : n ∈ N} is an element of B. Sf in (A, B) is the selection hypothesis: for each sequence (An : n ∈ N) of elements of A there isa sequence (Bn : n ∈ N) of finite sets such that for each n, Bn ⊆ An , and n∈N Bn ∈ B. In this paper, by a cover we mean a nontrivial one, that is, U is a cover of X if X = U and X ∈ / U. An open cover of a space is large if each element of the space belongs to infinitely many elements of the cover. An open cover U of a space X is called: • an ω-cover (a k-cover) if each finite (compact) subset C of X is contained in an element of U; • a γ-cover (a γk -cover) if U is infinite and for each finite (compact) subset C of X the set {U ∈ U : C U } is finite. Because of these definitions all spaces are assumed to be Hausdorff noncompact, unless otherwise stated. Let us mention that any ω-cover (k-cover) is infinite and large, and that any infinite subfamily of a γ-cover (γk -cover) is also a γ-cover (γk -cover). For a topological space X we denote: • O — the family of all open covers of X; • Γ — the family of all open γ-covers of X; • Γk — the family of all open γk -covers of X; • Ω — the family of all open ω-covers of X; • K — the family of all open k-covers of X. Different Δ-covers (k-covers, ω-covers, kF -covers, cF -covers,...) exposed many dualities in hyperspace topologies such as co-compact topology F+ , cofinite topology Z+ , Pixley-Roy topology, Fell topology and Vietoris topology. They also play important roles in selection principles ([1,4-9,12-16]). In this paper we continue to research relationships between closure-type properties of hyperspaces over a space X and covering properties of X. We 2
investigate selectors for sequence of subsets of the space 2X with the Z+ topology and the upper Fell topology (F+ -topology). Also we consider the selection properties of the bitopological space (2X , F+ , Z+ ). 2. Main definitions and notation The following lammas will be often used throughout the paper, sometimes without explicit reference. Lemma 2.1. (Lemma 1 in [1]) Let Y be an open subset of a space X and U an open cover of Y . Then the following holds: (1) U is a k-cover of Y ⇔ Y c ∈ ClF + (U c ); (2) U is an ω-cover of Y ⇔ Y c ∈ ClZ + (U c ). Lemma 2.2. (Lemma 2 in [1]) Let X be a topological space, Y an open subsets of X and U = {Un : n ∈ N} an open cover of Y . Then (1) U is a γk -cover of Y ⇔ the sequence (Unc : n ∈ N) converges to Y c in (2X , F+ ); (2) U is an γ-cover of Y ⇔ the sequence (Unc : n ∈ N) converges to Y c in X (2 , Z+ ). Lemma 2.3. (Lemma 3 in [1]) Given a space X and an open cover U of X the following holds: (1) U is a k-cover of X ⇔ U c is a dense subset of (2X , F+ ); (2) U is an ω-cover of X ⇔ U c is a dense subset of (2X , Z+ ). Let X be a topological space, and x ∈ X. A subset A of X converges to x, x = lim A, if A is infinite, x ∈ / A, and for each neighborhood U of x, A \ U is finite. Consider the following collection: • Ωx = {A ⊆ X : x ∈ A \ A}; • Γx = {A ⊆ X : x = lim A}. Note that if A ∈ Γx , then there exists {an } ⊂ A converging to x. So, simply Γx may be the set of non-trivial convergent sequences to x. Definition 2.4. Let X be a space and let U = {Uα : α ∈ Λ} be an open cover of X. Then U c = {Uαc : α ∈ Λ} converges to {∅} in (2X , τ ) where τ is a topology on 2X , if for every F ∈ 2X the U c converges to F , i.e. for each neighborhood W of F in the space (2X , τ ), |{α : Uαc W, α ∈ Λ}| < ℵ0 . Lemma 2.5. Let X be a space and let U = {Uα : α ∈ Λ} be an open cover of X. Then the following are equivalent: 3
1. U is an γ-cover of X; 2. U c converges to {∅} in (2X , Z+ ). For a topological space (2X , τ ) we denote: • DΩ — the family of dense subsets of (2X , τ ); • DΓ — the family of converging to {∅} subsets of (2X , τ ). Since every γ-cover contains a countably γ-cover, then each converging to {∅} subset of (2X , Z+ ) contains a countable converging to {∅} subset of (2X , Z+ ). 3. Hyperspace (2X , Z+ ) Theorem 3.1. Assume that Φ, Ψ ∈ {Γ, Ω}, ∈ {1, f in}. Then for a space X the following statements are equivalent: 1. X satisfies S (Φ, Ψ); 2. (2X , Z+ ) satisfies S (DΦ , DΨ ). Proof. We prove the theorem for = f in, the other proofs being similar. (1) ⇒ (2). Let (Di : i ∈ N) be a sequence of dense subsets of (2X , Z+ ) such that Di ∈ DΦ for each i ∈ N. Then (Dic : i ∈ N) is a sequence of open covers of X such that Dic ∈ Φ for each i ∈ N. Since X satisfies Sf in (Φ, Ψ), thereis a sequence (Ai : i ∈ N) offinite sets such that for each i, Ai ⊆ Dic , and i∈N Ai ∈ Ψ. It follows that i∈N Aci ∈ DΨ . (2) ⇒ (1). Let (Un : n ∈ N) be a sequence of open covers of X such that Un ∈ Φ. For each n, An := Unc is a dense subset of (2X , Z+ ) such that An ∈ DΦ . Applying that (2X , Z+ ) satisfies Sf in (DΦ , DΨ ), there is a sequence (An : n∈ N) of finite sets such that for each n, An ⊆ An , and n∈N An ∈ DΨ . Then n∈N Un is an open cover of X where Un = Acn for each n ∈ N and n∈N Un ∈ Ψ. Corollary 3.2. (Theorem 5 in [8]) For a space X the following are equivalent: 1. X satisfies S1 (Ω, Ω); 2. (2X , Z+ ) satisfies S1 (DΩ , DΩ ). Corollary 3.3. (Theorem 13 in [8]) For a space X the following are equivalent: 4
1. X satisfies Sf in (Ω, Ω); 2. (2X , Z+ ) satisfies Sf in (DΩ , DΩ ). Theorem 3.4. Assume that Φ, Ψ ∈ {Γ, Ω, }, ∈ {1, f in}. Then for a space X the following statements are equivalent: 1. Each open set Y ⊂ X has the property S (Φ, Ψ); 2. For each E ∈ 2X , (2X , Z+ ) satisfies S (ΦE , ΨE ). Proof. We prove the theorem for = 1, the other proofs being similar. (1) ⇒ (2). Let E ∈ 2X and let (An : n ∈ N) be a sequence such that An ∈ ΦE for each n ∈ N. Then (Acn : n ∈ N) is a sequence of open covers of E c such that Acn ∈ Φ for each n ∈ N. Since E c has the property S (Φ, Ψ), there is a sequence (Acn : n ∈ N) such that Acn ∈ Acn for each n ∈ N and {Acn : n ∈ N} is open cover of E c such that {Acn : n ∈ N} ∈ Ψ. It follows that {An : n ∈ N} ∈ ΨE . (2) ⇒ (1). Let Y be an open subset of X and let (Fn : n ∈ N) be a sequence of open covers of Y such that Fn ∈ ΦY (where ΦY is the Φ family of covers of Y ). Let E = X \Y . Put An = Fnc for each n ∈ N. Then An ⊂ 2X and An ∈ ΦE for each n ∈ N. Since, by (2), (2X , Z+ ) satisfies S1 (ΦE , ΨE ), there is a sequence (An : n ∈ N) such that An ∈ An for each n ∈ N and {An : n ∈ N} ∈ ΨE . It follows that {Fn : Fn = Acn , n ∈ N} ∈ Ψ. Corollary 3.5. (Theorem 3 in [5]) For a space X the following statements are equivalent: 1. Each open set Y ⊂ X has the property S1 (Ω, Γ); 2. (2X , Z+ ) is Fr´ echet-Urysohn; X + 3. (2 , Z ) is strongly Fr´ echet-Urysohn. Corollary 3.6. (Theorem 1 in [8]) For a space X the following are equivalent: 1. Each open set Y ⊂ X satisfies S1 (Ω, Ω); 2. (2X , Z+ ) has countable strong fan tightness (For each E ∈ 2X , (2X , Z+ ) satisfies S1 (ΩE , ΩE )). Corollary 3.7. (Theorem 9 in [8]) For a space X the following are equivalent: 1. Each open set Y ⊂ X satisfies Sf in (Ω, Ω); 5
2. (2X , Z+ ) has countable fan tightness (For each E ∈ 2X , (2X , Z+ ) satisfies Sf in (ΩE , ΩE )). Recall that a space is perfect if every open subset is an Fσ -subset [10]. Clearly every semi-stratifiable space is perfect. Note that all properties in the Scheepers Diagram ([3, 18]) are hereditary for Fσ -subsets (Corollary 2.4 in [11]). Proposition 3.8. Assume that Φ, Ψ ∈ {Γ, Ω, }, ∈ {1, f in}. Then for a perfect topological space X the following statements are equivalent: 1. X satisfies S (Φ, Ψ); 2. Each open set Y ⊂ X has the property S (Φ, Ψ). Theorem 3.9. Assume that Φ, Ψ ∈ {Γ, Ω}, ∈ {1, f in}. Then for a perfect space X the following statements are equivalent: 1. X satisfies S (Φ, Ψ); 2. (2X , Z+ ) satisfies S (DΦ , DΨ ); 3. For each E ∈ 2X , (2X , Z+ ) satisfies S (ΦE , ΨE ). Clearly that every perfectly normal space is perfect. Corollary 3.10. Assume that Φ, Ψ ∈ {Γ, Ω}, ∈ {1, f in}. Then for a perfectly normal space X the following statements are equivalent: 1. X satisfies S (Φ, Ψ); 2. (2X , Z+ ) satisfies S (DΦ , DΨ ); 3. For each E ∈ 2X , (2X , Z+ ) satisfies S (ΦE , ΨE ). 4. Hyperspace (2X , F+ ) Note that (2X , F+ ) satisfies the selection principle Sf in (O, O) for any space X, because (2X , F+ ) is always compact (see [2]). Sf in (O, O) property is called the Menger property (see [3, 18]). Lemma 4.1. Let X be a space and let U = {Uα : α ∈ Λ} be an open cover of X. Then the following are equivalent: 1. U is an γk -cover of X; 2. U c converges to {∅} in (2X , F+ ). 6
Theorem 4.2. Assume that Φ, Ψ ∈ {Γk , K}, ∈ {1, f in}. Then for a space X the following statements are equivalent: 1. X satisfies S (Φ, Ψ); 2. (2X , F+ ) satisfies S (DΦ , DΨ ). Proof. The proof is similar to the proof of Theorem 3.1. Corollary 4.3. (Theorem 4 in [8]) For a space X the following are equivalent: 1. X satisfies S1 (K, K); 2. (2X , F+ ) satisfies S1 (DΩ , DΩ ). Corollary 4.4. (Theorem 12 in [8]) For a space X the following are equivalent: 1. X satisfies Sf in (K, K); 2. (2X , F+ ) satisfies Sf in (DΩ , DΩ ). Theorem 4.5. Assume that Φ, Ψ ∈ {Γk , K}, ∈ {1, f in}. Then for a space X the following statements are equivalent: 1. Each open set Y ⊂ X has the property S (Φ, Ψ); 2. For each E ∈ 2X , (2X , F+ ) satisfies S (ΦE , ΨE ). Proof. The proof is similar to the proof of Theorem 3.4. Corollary 4.6. (Theorem 23 in [1]) For a space X the following are equivalent: 1. For each E ∈ 2X , (2X , F+ ) satisfies S1 (ΓE , ΩE ); 2. Each open set Y ⊂ X has the property S1 (Γk , K). Corollary 4.7. (Theorem 32 in [1]) For a space X the following are equivalent: 1. For each E ∈ 2X , (2X , F+ ) satisfies S1 (ΓE , ΓE ); 2. Each open set Y ⊂ X has the property S1 (Γk , Γk ).
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Definition 4.8. A subset A of a space X is called an k-Fσ -set if A can be ∞ Fi where Fi is a closed set in X for each i ∈ N and represented as A = i=1
for any compact set B ⊆ A there exists i ∈ N such that B ⊆ Fi . Definition 4.9. A space X is called k-perfect if every open subset is an k-Fσ -subset of X. Clearly that every k-perfect space is perfect. Proposition 4.10. Every perfectly normal space is k-perfect. Proof. Let U be an open subset of a perfectly normal space X. Then U ∞ may be represented as U = Fi where Fi is closed set of X for each i ∈ N i=1
and Fi ⊂ Fi+1 . Since X is a perfectly normal space, there exists a sequence (Ui : i ∈ N) of open sets of X such that Fi ⊆ Ui ⊆ Ui ⊆ U and Ui ⊆ Ui+1 for ∞ Ui and for any compact set B ⊆ U there each i ∈ N. It follow that U = exists i ∈ N such that B ⊆ Ui .
i=1
Note that, by Proposition 4.10, each cozero set is an k-Fσ -set. Question. Is there a k-perfect which is not (perfectly) normal space? Denote by S the Sorgenfrey line. Proposition 4.11. The space S2 is perfect, but not k-perfect. Proof. By Lemma 2.3 in [10], S2 is perfect. Consider the open set U = {(x, y) ∈ S2 : −x < y} {(x, −x) : x ∈ P} in the space S2 . ∞ Fi where Fi is closed Assume that U is a k-Fσ -set of S2 . Then U = i=1
set in S2 for each i ∈ N and for any compact set B ⊆ U there exists i ∈ N such that B ⊆ Fi . For each point1p = (y, 1−y) ∈ {(x, −x) : x ∈ P} we fix the compact set Zp = {p} {(y + n , −y + n ) : n ∈ N}. Note that Zp ⊂ U ∞ for each p ∈ {(x, −x) : x ∈ P}. Since U = Fi there exists k ∈ N such i=1
that |{p : Zp ⊂ Fk }| > ℵ0 . Since the set {p : Zp ⊂ Fk } is uncountable 8
subset of {(x, −x) : x ∈ P} there is a accumulation point z in subspace {(x, −x) : x ∈ R} of the space R2 . Clearly that for any neighborhood O(z) of z in the space S2 we have that O(z) Zp = ∅. It follows that Zp ⊂Fk O(z) Fk = ∅. Hence Fk is not closed set in S2 , a contradiction. Theorem 4.12. Assume that Φ, Ψ ∈ {Γk , K}, ∈ {1, f in}, X has the property S (Φ, Ψ) and A is an k-Fσ -set. Then A has the property S (Φ, Ψ). Proof. We prove the theorem for = f in, the other proofs being similar. Assume that X has the property Sf in (Φ, Ψ) and A is an k-Fσ -set. Consider a sequence (Ui : i ∈ N) of covers A such that Ui ∈ ΦA (where ΦA is the ∞ Fi where Fi is a closed Φ family of covers of A) for each i ∈ N. Let A = i=1
set in X for each i ∈ N and for any compact set B ⊆ A there exists i ∈ N such that B ⊆ Fi . Consider Vi = {(X \ Fi ) U : U ∈ Ui } for each i ∈ N. We claim that Vi ∈ Φ for each i ∈ N. Let S be a compact subset of X. Then U ∈ Ui such that hence there is S Fi is a compact subset of A and S Fi ⊂ U . It follows that S ⊂ (X \ Fi ) U for (X \ Fi ) U ∈ Vi . Since X has the property Sf in (Φ, Ψ), there is a sequence (Bi : i ∈ N) of finite sets such thatfor eachi, Bi ⊂ V , and i i∈N Bi ∈ Ψ. We claim that i∈N {Bi Fi : Bi Fi ⊂ Ui , i ∈ N} ∈ ΨA . Let B be a compact subset of A then there exists i ∈ N such that B ⊆ Fi . Since i∈N Bi is a large cover of X there is k ∈N and Vk ∈ Bk ⊂ Vk such that k > i and B ⊂ Vk . But Vk = (X \ Fk ) Uk for Uk ∈ Uk . Since k > i , B ⊂ Uk . It follows that A has the property Sf in (Φ, Ψ).
Proposition 4.13. Assume that Φ, Ψ ∈ {Γk , K}, ∈ {1, f in}. Then for a k-perfect space X the following statements are equivalent: 1. X satisfies S (Φ, Ψ); 2. Each open set Y ⊂ X has the property S (Φ, Ψ). By Proposition 4.10, we have the following result. Proposition 4.14. Assume that Φ, Ψ ∈ {Γk , K}, ∈ {1, f in}. Then for a perfectly normal space X the following statements are equivalent: 1. X satisfies S (Φ, Ψ); 9
2. Each open set Y ⊂ X has the property S (Φ, Ψ). Theorem 4.15. Assume that Φ, Ψ ∈ {Γk , K}, ∈ {1, f in}. Then for a k-perfect space X the following statements are equivalent: 1. X satisfies S (Φ, Ψ); 2. (2X , F+ ) satisfies S (DΦ , DΨ ); 3. For each E ∈ 2X , (2X , F+ ) satisfies S (ΦE , ΨE ). By Proposition 4.10, we have the following result. Corollary 4.16. Assume that Φ, Ψ ∈ {Γk , K}, ∈ {1, f in}. Then for a perfectly normal space X the following statements are equivalent: 1. X satisfies S (Φ, Ψ); 2. (2X , F+ ) satisfies S (DΦ , DΨ ); 3. For each E ∈ 2X , (2X , F+ ) satisfies S (ΦE , ΨE ). 5. Bitopological space (2X , F+ , Z+ ) Theorem 5.1. Assume that Φ ∈ {Γk , K}, Ψ ∈ {Γ, Ω}, ∈ {1, f in}. Then for a space X the following statements are equivalent: 1. X satisfies S (Φ, Ψ); + Z+ ). 2. (2X , F+ , Z+ ) satisfies S (DΦF , DΨ Proof. The proof is similar to the proof of Theorem 3.1. Corollary 5.2. (Theorem 6 in [8]) For a space X the following are equivalent: 1. X satisfies S1 (K, Ω); + + 2. (2X , F+ , Z+ ) satisfies S1 (DΩF , DΩZ ). Corollary 5.3. (Theorem 14 in [8]) For a space X the following are equivalent: 1. X satisfies Sf in (K, Ω); + + 2. (2X , F+ , Z+ ) satisfies Sf in (DΩF , DΩZ ). Theorem 5.4. Assume that Φ ∈ {Γk , K}, Ψ ∈ {Γ, Ω}, ∈ {1, f in}. Then for a space X the following statements are equivalent: 10
1. Each open set Y ⊂ X has the property S (Φ, Ψ); + Z+ 2. For each E ∈ 2X , (2X , F+ , Z+ ) satisfies S (ΦF E , ΨE ). Proof. The proof is similar to the proof of Theorem 3.4. Corollary 5.5. (Theorem 31 in [1]) For a space X the following are equivalent: 1. Each open set Y ⊂ X has the property S1 (Γk , Ω); + Z+ 2. For each E ∈ 2X , (2X , F+ , Z+ ) satisfies S1 (ΓF E , ΩE ). Corollary 5.6. (Theorem 33 in [1]) For a space X the following are equivalent: 1. Each open set Y ⊂ X has the property S1 (Γk , Γ); + Z+ 2. For each E ∈ 2X , (2X , F+ , Z+ ) satisfies S1 (ΓF E , ΓE ). Theorem 5.7. Assume that Φ ∈ {Γk , K}, Ψ ∈ {Γ, Ω}, ∈ {1, f in}, X has the property S (Φ, Ψ) and A is an k-Fσ -set. Then A has the property S (Φ, Ψ). Proof. The proof is similar to the proof of Theorem 4.12. Proposition 5.8. Assume that Φ ∈ {Γk , K}, Ψ ∈ {Γ, Ω}, ∈ {1, f in}. Then for a k-perfect space X the following statements are equivalent: 1. X satisfies S (Φ, Ψ); 2. Each open set Y ⊂ X has the property S (Φ, Ψ). Theorem 5.9. Assume that Φ ∈ {Γk , K}, Ψ ∈ {Γ, Ω}, ∈ {1, f in}. Then for a k-perfect space X the following statements are equivalent: 1. X satisfies S (Φ, Ψ); 2. (2X , F+ , Z+ ) satisfies S (DΦ , DΨ ); 3. For each E ∈ 2X , (2X , F+ , Z+ ) satisfies S (ΦE , ΨE ).
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