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Second Workshop Coverings, Selections and Games in Topology Lecce, Italy, December 19–22, 2005 Convergence properties of hyperspaces Giuseppe Di Maio Seconda Universita di Napoli, Caserta, Italy [email protected] 2X the set of closed subsets of X ∈ T2 A ⊂ X, A a family of subsets of X: Ac = X \ A and Ac = {Ac : A ∈ A}, A− = {F ∈ 2X : F ∩ A 6= ∅}, A+ = {F ∈ 2X : F ⊂ A}. ∆ ⊂ 2X closed for finite unions and containing all singletons upper ∆-topology ∆+ has a base {(Dc)+ : D ∈ ∆} ∪ {2X }. 1. ∆ = CL(X): upper Vietoris topology V+ 2. ∆ = K(X): upper Fell topology F+ 3. ∆ = F(X): Z+-topology The lower Vietoris topology V− is generated by all the sets U −, U ⊂ X open. The ∆-topology : τ∆ = ∆+ ∨ V−. sets: τ∆-basic (Dc)+∩(∩i≤mVi−), D ∈ ∆, V1, · · · , Vm open in X. ∆-topologies : the Vietoris topology V = V+ ∨ V− the Fell topology F = F+ ∨ V− the topology Z = Z+ ∨ V− Let A and B be sets whose elements are families of subsets of an infinite set X. Then: S1(A, B) denotes the selection principle: 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) denotes the selection hypothesis: For each sequence (An : n ∈ N) of elements of A there is a sequence (Bn : n ∈ N) of finite sets such that for each S n Bn ⊂ An and n∈N Bn is an element of B. For a space X, x ∈ X, ∆ ⊂ 2X : • O: the collection of open covers of X; • Ω: the collection of ω-covers of X; • K: the collection of k-covers of X; • Γ: the collection of γ-covers; • Γk : the collection of γk -covers; • Γ∆: the collection of γ∆-covers; • Ωx: the set {A ⊂ X \ {x} : x ∈ A}; • Σx: the set of all nontrivial sequences in X that converge to x. Let us recall that if ∆ ⊂ 2X , then an open cover U of X is called a ∆-cover if each D ∈ ∆ is contained in an element of U and X does not belong to U (i.e. the cover is not trivial). F(X)-covers (resp. K(X)-covers) are called ωcovers (resp. k-covers). An open cover U of X is said to be a γ∆-cover if it is infinite and for each D ∈ ∆ the set {U ∈ U : D * U } is finite. γF(X)-covers (resp. γK(X)-covers) are called γcovers (resp. γk -covers. Observe that each infinite subset of a γ∆-cover is still a γ∆-cover. So, we may suppose that such covers are countable. αi-properties in hyperspaces A space X has property αi, i = 1, 2, 3, 4, if for each x ∈ X and each sequence (σn : n ∈ N) of elements of Σx there is a σ ∈ Σx such that: α1: for each n ∈ N the set σn \ σ is finite; α2: for each n ∈ N the set σn ∩ σ is infinite; α3: for infinitely many n ∈ N the set σn ∩ σ is infinite; α4: for infinitely many n ∈ N the set σn ∩ σ is nonempty. Evidently, α1 ⇒ α2 ⇒ α3 ⇒ α4. Theorem 1 For a space X and a collection ∆ ⊂ 2X the following statements are equivalent: (1) (2X , ∆+) is an α2-space; (2) (2X , ∆+) is an α3-space; (3) (2X , ∆+) is an α4-space; (4) For each E ∈ 2X , (2X , ∆+) satisfies S1(ΣE , ΣE ); (5) Each open set Y ⊂ X satisfies S1(Γ∆, Γ∆). Remark 2 The statement (1) implies (2): (1) For each E ∈ 2X , (2X , τ∆) satisfies S1(ΣE , ΣE ); (2) Each open set Y ⊂ X satisfies S1(Γ∆, Γ∆) (equivalently, Sf in(Γ∆, Γ∆)). Two consequences of Theorem 1. Corollary 3 For a space X TFAE: (1) (2X , F+) is an α4-space; (2) Each open set Y ⊂ X is an S1(Γk , Γk )-set. Corollary 4 For a space X TFAE: (1) (2X , Z+) is an α4-space; (2) Each open set Y ⊂ X is an S1(Γ, Γ)-set. A space is said to be a γ-set (resp. γk -set) if it satisfies the selection principle S1(Ω, Γ) (resp. S1(K, Γk )). (2X , Z+) is FU ((2X , F+) is SFU) if and only if each open set Y ⊂ X is a γ-set (a γk -set). Γ ⊂ Ω and Γk ⊂ K, it follows that S1(Ω, Γ) ⊂ S1(Γ, Γ) and S1(K, Γk ) ⊂ S1(Γk , Γk ). So: Proposition 5 For a space X the following statements hold: (1) If (2X , Z+) is a Fréchet-Urysohn space, then it is an α2-space; (2) If (2X , F+) is strongly Fréchet-Urysohn, then it is an α2-space. Theorem 6 For a space X and a collection ∆ ⊂ 2X the following are equivalent: (1) (2X , ∆+) is an α1-space; (2) For each open set Y ⊂ X and each sequence (Un : n ∈ N) of γ∆-covers of Y there is a γ∆-cover U of Y intersecting each Un in all but finitely many elements. FU-type properties X is FU if ∀x ∈ X and ∀A ∈ Ωx ∃ a sequence (xn : n ∈ N) in A belonging to Σx. X is sequential if for each non-closed set A ⊂ X there are a point x ∈ X \A and a sequence (xn : n ∈ N) in A that belongs to Σx. X has countable tightness if for each x ∈ X and each A ∈ Ωx there is a countable element B ∈ Ωx such that B ⊂ A. A space X has the Reznichenko property if for each x ∈ X and each A ∈ Ωx there is a sequence (Bn : n ∈ N) of pairwise disjoint, finite subsets of A such that each neighborhood U of x intersects all but finitely many sets Bn. A space X has the Pytkeev property if for each x ∈ X and each A ∈ Ωx there is a sequence (Bn : n ∈ N) of infinite, countable subsets of A such that each neighborhood U of x contains some Bn. A space X is said to be: FF: filter-Fréchet if for each x ∈ X and each A ∈ Ωx there is a sequence (Fn : n ∈ N) of filter-bases on A such that: (FF1) For each n ∈ N, there is an Fn ∈ Fn such that x ∈ / Fn; (FF2) For each neighborhood U of x there is n0 ∈ N such that n ≥ n0 implies Fn ⊂ U for some Fn ∈ Fn. SFF: strongly filter-Fréchet if for each x ∈ X and each A ∈ Ωx there is a sequence (Fn : n ∈ N) of filter-bases on A satisfying (FF1) and (FF2) above and the condition (FF3) For each n ∈ N there is a countable F ∈ Fn . SSF: strongly set-Fréchet if for each x ∈ X and each A ∈ Ωx there is a sequence (Bn : n ∈ N) of pairwise disjoint subsets of A such that the following conditions hold: (SF1) x ∈ / Bn for each n ∈ N; (SF2) each neighborhood U of x intersects all but finitely many sets Bn; (SF3) each Bn is countable. SF: set-Fréchet if only conditions (SF1) and (SF2) in SSF are satisfied. CT µ 6 ¡ ¡ compact ¡ ¡ FU SFF SSF - - T2 SF¾ FF¾ SFF - 6 ? Seq Pyt - Rez - Theorem 7 For a space X and a family ∆ ⊂ 2X the following statements are equivalent: (1) (2X , ∆+) is a filter-Fréchet space; (2) For each open set Y ⊂ X and each ∆-cover U of Y there is a sequence (Bn : n ∈ N) of filter-bases on U such that: (i) For each n, there is Cn ∈ Bn which is not a ∆-cover of Y ; (ii) For each D ∈ ∆ with D ⊂ Y there is n0 ∈ N such that whenever n ≥ n0, then there exists Hn ∈ Bn satisfying D ⊂ H for every H ∈ Hn. Theorem 8 For a space X the following statements are equivalent: (1) (2X , ∆+) is a strongly filter-Fréchet space; (2) For each open set Y ⊂ X and each ∆-cover U of Y there is a sequence (Bn : n ∈ N) of filter-bases on U such that: (i) For each n, there is Cn ∈ Bn which is not a ∆-cover of Y ; (ii) For each D ∈ ∆ with D ⊂ Y there is n0 ∈ N such that whenever n ≥ n0, then there exists Hn ∈ Bn satisfying D ⊂ H for every H ∈ Hn; (iii) For each n ∈ N there is some countable element in Bn. The implication (1) ⇒ (2) in Theorem 7 and in Theorem 8 is still valid if the ∆+-topology is replaced with the τ∆-topology. Example 9 [CH] There exists a space X satisfying the condition (2) in the previous theorem, but (2X , τ∆) is not a strongly filter-Fréchet space. Let X be the Hausdorff, compact, hereditarily Lindelöf, non hereditarily separable space constructed by Kunene, and let ∆ = K(X), hence τ∆ = F. (CL(X), F) has uncountable tightness so that it is not SFF. Let us show that the condition (2) in Theorem 8 holds. Let Y be any open subset of X and let U be a k-cover of Y in X. As Y is locally compact and Lindelöf, it is hemicompact. Let (Kn : n ∈ N) be an increasing countable family of compact subsets of Y such that each compact subset of Y is contained in some Kn. For each n pick a set Un ∈ U such that Kn ⊂ Un. Since U is a k-cover of Y in X, the family {Y ∩ U : U ∈ U } is a k-cover of Y in Y , so that Y is not a member of {Y ∩ U : U ∈ U}. Thus for each n ∈ {1, · · · , n0}, where n0 is some element in N, pick a point xn ∈ Y \ Un and for each n ∈ N define Bn = {{Ui : n ≤ i ≤ n∗} : n∗ ≥ n}. It is clear that {Ui : n ≤ i ≤ n∗1} ⊂ {Ui : n ≤ i ≤ n∗2} for n ≤ n∗1 ≤ n∗2, so that the collection Bn is linearly ordered by inclusion and in particular it is a filter base. We show that the sequence (Bn : n ∈ N) satisfies: (i) For each n, there is Cn ∈ Bn which is not a k-cover of Y ; (ii) For each compact set K ⊂ Y there is n0 ∈ N such that whenever n ≥ n0, then there exists Hn ∈ Bn satisfying K ⊂ H for every H ∈ Hn; (iii) For each n ∈ N there is some countable element in Bn. Of course, (iii) is satisfied because every element of Bn is finite. The condition (i) is also true. Indeed, for a given i ∈ N, we have by the choice of points xn that no element of {Ui : n ≤ i ≤ n∗} includes the set {xi : n ≤ i ≤ n∗} which is a compact subset of Y . Finally, let us prove that (ii) holds. Let K be a compact subset of Y . There exists n0 ∈ N such that K ⊂ Kn0 . For each n ≥ n0 take any element Hn := {Ui : n ≤ i ≤ n} in Bn = {{Ui : n ≤ i ≤ n∗} : n∗ ≥ n}, where n is an element of N with n ≥ n. Then we have K ⊂ H for each H ∈ Hn. Indeed, for each i ∈ N with n ≤ i ≤ n we have K ⊂ Kn0 ⊂ Kn ⊂ Ki ⊂ Ui. ¤ Theorem 10 For a space X the following statements are equivalent: (1) (2X , ∆+) has the strong set-Fréchet property; (2) For each open set Y ⊂ X and each ∆-cover U of Y there is a sequence (Vn : n ∈ N) of countable, pairwise disjoint subsets of U such that: (i) no Vn is a ∆-cover of Y ; (ii) each D ∈ ∆ which is a subset of Y is contained in an element of Vn for all but finitely many n. Theorem 11 For a space X the following statements are equivalent: (1) (2X , ∆+) is a set-Fréchet space; (2) For each open set Y ⊂ X and each ∆-cover U of Y there is a sequence (Vn : n ∈ N) of pairwise disjoint subsets of U such that: (i) no Vn is a ∆-cover of Y ; (ii) each D ⊂ Y such that D ∈ ∆ is contained in an element of Vn for all but finitely many n. References • A.V. Arhangel’skiǐ, The frequency spectrum of a topological space and the classification of spaces, Soviet Math. Doklady 13 (1972), 1185–1189. • A.V. Arhangel’skiǐand T. Nogura, Relative sequentiality, Topology Appl. 82 (1998), 49–58. • A. Caserta, G. Di Maio, Lj.D.R. Kočinac, E. MECCARIELLO, Applications of k-covers II, Topology Appl., accepted. • C. Costantini, Ĺ. Holá and P. Vitolo, Tightness, character and related properties of hyperspace topologies, Topology Appl. 142 (2004), 245–292. • G. Di Maio, Lj.D.R. Kočinac, E. MECCARIELLO, Selection principles and hyperspace topologies, Topology Appl., to appear. • G. Di Maio, E. MECCARIELLO, S. Naimpally, Uniformizing (proximal) ∆-topologies, Topology Appl. 137 (2004), 99–113. • K. Kunen, A compact L-space under CH, Topology Appl. 12 (1981), 283–287.