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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.
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FU SFF SSF
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SF¾ FF¾ SFF
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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.
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