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Fell continuous multi-sections
Topology and its Applications 160 (2013) 906–914
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Topology and its Applications www.elsevier.com/locate/topol
Fell continuous multi-sections David Buhagiar ∗ , Valentin Gutev Department of Mathematics, Faculty of Science, University of Malta, Msida, MSD 2080, Malta
a r t i c l e
i n f o
Article history: Received 19 October 2012 Received in revised form 5 March 2013 Accepted 6 March 2013 MSC: 54C60 54C65 54B20 54D15
a b s t r a c t The paper is devoted to several characterisations of metacompact-like properties in terms of special sections continuous with respect to the Fell hyperspace topology. These results fit naturally within the framework of covering properties described by similar sections, but continuous with respect to other hyperspace topologies. © 2013 Elsevier B.V. All rights reserved.
Keywords: Set-valued mapping Hyperspace Vietoris topology Fell topology Lower semi-continuous Closed-graph Section F -ultranormal space
1. Introduction All spaces in this paper are assumed to be T 1 -spaces. For space Y , let 2Y be the power set of Y ; F (Y ) be the set of all nonempty closed subsets of Y ; and C (Y ) — that of all compact members of F (Y ). Recall that the Vietoris topology τ V on F (Y ) is generated by all collections of the form
V = S ∈ F (Y ): S ⊂ V and S ∩ V = ∅, whenever V ∈ V , where V runs over the finite families of open subsets of Y . Another topology on F (Y ) which will play an important role in this paper is the Fell topology τ F , it is generated by all basic Vietoris neighbourhoods V such that V is a finite family of open subsets of Y and Y \ V is compact. In what follows, let us make the explicit agreement that when the clarity seems to demand it, for a given topology τ on F (Y ) and Q ⊂ F (Y ), we will broadly use (Q , τ ) to express that Q is endowed with τ . For instance, we will often write ϕ : X → (C (Y ), τ ) for a map from X to the space (C (Y ), τ ); F (C (Y ), τ ) will mean the nonempty closed subsets of (C (Y ), τ ); etc. In particular, we shall say that a set-valued mapping ϕ : X → F (Y ) is τ -continuous if it is continuous as a usual map from X to the space (F (Y ), τ ). Here are some natural examples of τ -continuous mappings. A mapping
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Corresponding author. E-mail addresses: [email protected] (D. Buhagiar), [email protected] (V. Gutev).
0166-8641/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.topol.2013.03.001
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ϕ : X → 2Y is lower semi-continuous, or l.s.c., if the set ϕ −1 [U ] = {x ∈ X: ϕ (x) ∩ U = ∅} is open in X for every open U ⊂ Y ; and ϕ is upper semi-continuous, or u.s.c., if ϕ −1 [ F ] is closed in X for every closed F ⊂ Y . Similarly, we will say that ϕ : X → F (Y ) is τ F -upper semi-continuous, or τ F -u.s.c., if ϕ −1 [ K ] is closed for every compact K ⊂ Y . We now have that ϕ : X → F (Y ) is τ V -continuous if and only if it is both l.s.c. and u.s.c.; and that ϕ is τ F -continuous if and only if it is both l.s.c. and τ F -u.s.c. There is a natural relationship between covering properties of topological spaces and selection type properties of setvalued mappings. Turning to this, let us recall that a map f : X → Y is a selection (or, a single-valued selection) for Φ : X → 2Y if f (x) ∈ Φ(x) for every x ∈ X . A set-valued mapping ϕ : X → 2Y is a multi-selection (or, a set-valued selection) for Φ : X → 2Y if ϕ (x) ⊂ Φ(x) for every x ∈ X ; and ϕ : X → 2Y is a section for Φ : X → 2Y if ϕ (x) ∩ Φ(x) = ∅ for every x ∈ X . If ϕ is a section for Φ , then both ϕ and Φ must be nonempty-valued. Of course, every nonempty-valued multi-selection for Φ is also a section for Φ . A space X is ultranormal if every two disjoint closed subsets are contained in disjoint clopen subsets, i.e. if the large inductive dimension of X is 0. In the realm of normal spaces, X is ultranormal if and only if its covering dimension is 0. The ultranormal paracompact spaces are often called ultraparacompact. As is well known, a Hausdorff space X is ultraparacompact if and only if every open cover of X has a pairwise disjoint open refinement, see [8]. The existence of such refinements is behind the following result. Theorem 1.1. A space X is ultraparacompact if and only if for every completely metrizable space Y , every l.s.c. mapping Φ : X → F (Y ) has a continuous selection (in particular, a τ V -continuous multi-selection). Theorem 1.1 is due to E. Michael [19, Theorem 2] (see also [18]) and T. Ishii [15, Theorem 3]. Using a classical result of Lefshetz [16] (see also [9, Theorem 1.5.18]), it follows that a space X is ultranormal if and only if for every point-finite open cover U of X there exists a clopen (point-finite) cover { V U : U ∈ U } of X such that V U ⊂ U , for all U ∈ U . In particular, X is an ultranormal metacompact space if and only if for every open cover U of X there exists a clopen point-finite cover { V U : U ∈ U } of X such that V U ⊂ U , for all U ∈ U . In view of this covering property, it makes sense to say that a space X is ultrametacompact if it is an ultranormal metacompact space. In slightly different terms (see, Proposition 3.2), the following result was obtained in [12, Theorem 5.2], see also [6, Theorem 6.1] and [12, Theorem 6.3]. Theorem 1.2. A space X is ultranormal if and only if for every metrizable space Y , every l.s.c. mapping Φ : X → C (Y ) has a τ F continuous multi-selection ϕ : X → C (Y ). In particular, X is ultrametacompact if and only if for every completely metrizable space Y , every l.s.c. mapping Φ : X → F (Y ) has a τ F -continuous multi-selection ϕ : X → C (Y ). Let W be a collection of subsets of a set X . If U , V ∈ W , then a finite sequence W 1 , W 2 , . . . , W k of elements of W is called a chain from U to V if U = W 1 , V = W k and W i ∩ W i +1 = ∅ for every i = 1, . . . , k − 1. A subset P ⊂ W is called connected if every pair of elements of P is connected by a chain. The components of W are defined as the maximal connected subsets of W . A space X is called superparacompact (Pasynkov, see [21]) if every open cover of X has an open finite component (i.e., having finite components) refinement. The following theorem was proved in [1]. Theorem 1.3. A space X is superparacompact if and only if for every completely metrizable space Y , every l.s.c. mapping Φ : X → F (Y ) has a τ V -continuous section ϕ : X → C (Y ).
W ∈ U for every finite W ⊂ U . For a collection U , let U F be the collection A cover U of X is finitely additive if of the unions of all finite subcollections of U . In these terms, a cover U of X is finitely additive if and only if U = U F . According to [2, Proposition 2.3] (see also [3, Theorem 2.2]), a space X is superparacompact if and only if every finitely additive cover U of X has a pairwise disjoint open refinement. In particular, a space X is superparacompact if and only if for every open cover U of X , the cover U F has a disjoint open refinement. Hence, superparacompact spaces can be regarded as “F -ultraparacompact”. This also reveals a natural relationship between Theorems 1.1 and 1.3, and the role of finitely additive covers to transform multi-selections into compact-valued sections. Motivated by this, we will say that a space X is F -ultranormal if for every point-finite open cover U of X , there exists a point-finite clopen cover { V W : W ∈ U F } of X with V W ⊂ W , for all W ∈ U F . The following finitely additive version of Theorem 1.2 will be proved in this paper. Theorem 1.4. A space X is F -ultranormal if and only if for every completely metrizable space Y , every l.s.c. mapping Φ : X → C (Y ) admits a τ F -continuous mapping ψ : X → C (C (Y ), τ V ) such that K ∩ Φ(x) = ∅ for every K ∈ ψ(x) and x ∈ X . Let us explicitly remark that “C (C (Y ), τ V )” is the set of all nonempty compact subsets of the space (C (Y ), τ V ). That is, ψ is a map from X to the Fell hyperspace of the Vietoris hyperspace. Motivated by Theorem 1.4, we shall say that a mapping ψ : X → 2C (Y ) is a multi-section for Φ : X → 2Y if K ∩ Φ(x) = ∅ for every K ∈ ψ(x) and x ∈ X . The interested reader is referred to Section 4, where we discuss the subtle difference between sections and multi-sections. Just like before, we will say that a space X is F -ultrametacompact if it is an F -ultranormal metacompact space. The following consequence now follows from Theorem 1.4.
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Corollary 1.5. A space X is F -ultrametacompact if and only if for every completely metrizable space Y , every l.s.c. mapping Φ : X → F (Y ) admits a τ F -continuous multi-section ψ : X → C (C (Y ), τ V ). A space X is called supermetacompact [3, Definition 3.1] if every finitely additive open cover of X has a point-finite clopen refinement. Every ultrametacompact space is F -ultrametacompact, and every F -ultrametacompact space is supermetacompact, but neither of these is invertible. Indeed, every compact space is F -ultrametacompact, but there are compact spaces which are not ultranormal, for instance the closed unit interval. According to Example 3.7, there exists a supermetacompact space which is not F -ultrametacompact. Concerning multi-sections, we were able to provide a similar characterisation of supermetacompactness when the set-valued mappings take values in locally compact spaces, see Theorem 4.5. In general, for a completely metrizable range, we have only a partial result for these spaces, see Theorem 4.4. The proof of Theorem 1.4 is finalised in Section 3 (see, Theorem 3.3), while Corollary 1.5 is a partial case of a slightly more general result (see, Corollary 3.6). The proof of Theorem 1.4 is reduced to the existence of multi-selections for special star-mappings associated to set-valued mappings (Proposition 3.1). These multi-selections are constructed using special additive sieves which are developed in the next section. Another important ingredient in this proof is played by closedgraph mappings which are used as an interface to τ F -continuity, see Proposition 3.2. 2. Special additive sieves First of all, we need some standard terminology for set-valued mappings. Given Φ : X → 2Y , the inverse Φ −1 : Y → 2 X of Φ isdefined by Φ −1 ( y ) = Φ −1 [{ y }], y ∈ Y . In particular, we always have that (Φ −1 )−1 = Φ . Whenever S ⊂ X , let Φ[ S ] = {Φ(x): x ∈ S } which is a subset of Y and will play the role of the image of S by Φ , when Φ is considered as a set-valued mapping. Let us explicitly remark that we will also use Φ( S ) = {Φ(x): x ∈ S } which is a subset of 2Y , in fact it is the image of S by Φ , when Φ is considered as a usual map from X to the set 2Y . To every mapping Φ : X → 2Y we will associate another one Φ : X → 2Y defined by Φ(x) = Φ(x), x ∈ X . The mapping Φ will play the role of a point-wise closure of Φ . We now turn to some basic terminology for sieves. A partially ordered set ( T , ) is a tree if {s ∈ T : s t } is well-ordered for every t ∈ T . In this case, we use T (0) to denote the set of the minimal elements of T . Also, given an ordinal α , if T (β) is defined for every β < α , then T (α ) denotes the minimal elements of T \ ( T α ), where T α = { T (β): β < α }. The set T (α ) is called the α th-level of T , while the height of T is the least ordinal α such that T α = T . A maximal linearly ordered subset of T is called a branch, and B ( T ) is used to denote the set of all branches of T . Finally, for t ∈ T , the node of t in T is the subset node T (t ) ⊂ T of all immediate successors of t; for convenience, we also set node T (∅) = T (0). A tree ( T , ) is pruned if every element of T has a successor in T , i.e. node T (t ) = ∅ for every t ∈ T . Given a set Y and a pruned tree ( T , ) of height ω , a mapping S : T → 2Y is a sieve on Y if Y = S [node T (∅)] and S (t ) = S [nodeT (t )] forevery t ∈ T . For a tree ( T , ) and S : T → 2Y , the polar mapping ΩS : B ( T ) → 2Y associated to S −1 is defined by ΩS (β) = {S (t ): t ∈ β}, β ∈ B ( T ); and the inverse polar mapping 0S : Y → 2B( T ) is defined by 0S = ΩS . Following [5], a nonempty-open-valued sieve S : T → 2Y on a space Y is called a (λ)-sieve if for every β ∈ B ( T ) there exists an y (β) ∈ Y such that if U is a neighbourhood of y (β), then S (t ) ⊂ U for some t ∈ β . This clearly implies that y (β) ∈ S (t ) = S (t ) for every t ∈ β . Hence, for a Hausdorff space Y , we have that ΩS (β) = { y (β)} and, therefore, the polar mapping ΩS : B ( T ) → 2Y is singleton-valued. Since every singleton-valued mapping has the same graph as a singlevalued one (representing the same relation), we will make no difference between such mappings. Thus, in this case, the polar mapping ΩS : B ( T ) → Y is a usual map. A sieve S : T → 2Y on a space Y is locally-finite (point-finite, etc.) if each cover {S (t ): t ∈ T (n)}, n < ω , is locally-finite (point-finite, etc.). Let us explicitly mention the well-known fact that every complete metric space (Y , ρ ) has a nonemptyopen-valued locally-finite (λ)-sieve. For instance, use [20, Lemma 2.2] to get a nonempty-open-valued locally-finite sieve S : T → 2Y such that diamρ (S (t )) < 2−n for every t ∈ T (n) and n < ω . Then, S is a (λ)-sieve by Cantor’s intersection theorem. Following Nyikos [22], for a tree ( T , ) and t ∈ T , let
O (t ) = β ∈ B ( T ): t ∈ β .
If T is pruned and of height ω , the family {O (t ): t ∈ T } is a base for a completely metrizable (non-Archimedean) topology on B ( T ). We will refer to this topology as the branch topology, and to the resulting topological space as the branch space. Throughout this paper, B ( T ) will be always endowed with the branch topology when it comes to consider it as a topological space. It is well known that B ( T ) is compact if and only if all levels of T are finite. For a set D and a cardinal number κ ω , let [ D ]<κ = { S ⊂ D: 1 | S | < κ }. Following [13], for a pruned tree ( T , ) of height ω , we consider another tree (Σ T , ) defined in the following way. Let Σ T = {[ T (n)]<ω : n < ω}. Next, define a relation on Σ T by letting for σ , μ ∈ Σ T that σ ≺ μ if
μ⊂
node T (s): s ∈ σ
and
μ ∩ nodeT (s) = ∅, s ∈ σ .
(2.1)
Finally, extend this relation to a partial order on Σ T making it transitive. Thus, we get a pruned tree (Σ T , ) of height because so is ( T , ).
ω
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Proposition 2.1. Let (Y , ρ ) be a complete metric space, and M : T → 2Y be a nonempty-open-valued locally-finite sieve on Y such that diamρ (M (t )) < 2−n , for every t ∈ T (n) and n < ω . Define a mapping R : Σ T → 2C (Y ) by letting R (σ ) = M (σ ), for every σ ∈ ΣT . Then, R is an open-valued locally-finite (λ)-sieve on (C (Y ), τ V ). In particular, ΩR : B (ΣT ) → (C (Y ), τ V ) is a continuous perfect map. Proof. The fact that R is a sieve on C (Y ) follows by (2.1). According to the definition of R , we also have that it is openvalued and locally-finite. It is well known that the Vietoris topology on C (Y ) is metrizable with respect to the Hausdorff metric H (ρ ) on C (Y ) associated to ρ :
H (ρ )( S , T ) = max
ρ ( S , y ) + ρ ( y , T ): y ∈ S ∪ T ,
S , T ∈ C (Y ).
Since ρ is complete, so is the Hausdorff metric H (ρ ) on C (Y ). If σ ∈ Σ T , then σ belongs to the nth-level of the tree Σ T for some n < ω . We now have that H (ρ )( K 1 , K 2 ) < 2−n for every K 1 , K 2 ∈ R (σ ) = {M (t ): t ∈ σ } because diamρ (M (t )) < 2−n for all t ∈ σ . That is, diam H (ρ ) (R (σ )) 2−n and, by Cantor’s intersection theorem, R is a (λ)-sieve on C (Y ). Hence,
ΩR : B (Σ T ) → (C (Y ), τ V ) is a continuous map, see [11, Proposition 5.2]. Since R is a locally-finite closed-valued sieve 2
on C (Y ), the inverse 0R = Ω −1 is compact-valued and u.s.c. [11, Lemma 5.3], therefore ΩR is also perfect. R
To every mapping Φ : X → F (Y ) we are going to associate another one st[Φ, C ] : X → 2C (Y ) by letting for x ∈ X that
st[Φ, C ](x) = K ∈ C (Y ): K ∩ Φ(x) = ∅ .
(2.2)
The mapping st[Φ, C ] will play the role of the star of Φ with respect to the compact subsets of Y . The following propositions collect some basic properties of st[Φ, C ]; the verification of the first one is left to the reader. Proposition 2.2. If Φ : X → F (Y ) and C (Y ) is endowed with the Vietoris topology, then st[Φ, C ] : X → F (C (Y ), τ V ) and it is l.s.c. if and only if Φ is l.s.c. Proposition 2.3. Let Y be a metrizable space, M : T → 2Y be a nonempty-open-valued locally-finite sieve on Y , and let R : Σ T → 2C (Y ) be the sieve on (C (Y ), τ V ) defined by R (σ ) = M (σ ), for every σ ∈ Σ T . Also, let Φ : X → F (Y ) be an l.s.c. mapping. Define S : Σ T → 2 X by S (σ ) = (st[Φ, C ])−1 [R (σ )], σ ∈ Σ T . Then, S is an open-valued sieve on X such that
S (σ ) =
Φ −1 M (t ) : t ∈ σ ,
for every σ ∈ Σ T .
Proof. According to Proposition 2.2, S is an open-valued sieve on X . If σ ∈ Σ T and st[Φ, C ](x) ∩ R (σ ) = ∅, then there exists K ∈ C (Y ) such that K ∩ Φ(x) = ∅ and K ∈ {M (t ): t ∈ σ }, hence x ∈ {Φ −1 [M (t )]: t ∈ σ }. Conversely, if Φ(x) ∩ M (t ) = ∅ for some t ∈ σ , then there is a finite set F ⊂ Y such that F ∈ {M (t ): t ∈ σ } and F ∩ Φ(x) = ∅, i.e. x ∈ S (σ ) = (st[Φ, C ])−1 [R (σ )]. Thus, S (σ ) = {Φ −1 [M (t )]: t ∈ σ }, σ ∈ Σ T . 2 3. Metacompact-like spaces and multi-sections Recall that a mapping ψ : X → 2C (Y ) is a multi-section for Φ : X → 2Y if K ∩ Φ(x) = ∅, for every K ∈ ψ(x) and x ∈ X . Multi-sections are in good accordance with the star-mapping st[Φ, C ] defined in the previous section, see (2.2). Namely, the following observation follows immediately from the definitions. Proposition 3.1. For mappings Φ : X → F (Y ) and ψ : X → C (C (Y ), τ V ), the following are equivalent: (a) ψ is a multi-section for Φ . (b) ψ is a multi-selection for st[Φ, C ]. A mapping ψ : X → 2Y has a closed graph, or is a closed-graph mapping, if its graph Graph(ψ) = {(x, y ) ∈ X × Y : y ∈ ψ(x)}, is closed in X × Y . Proposition 3.2. Every closed-graph mapping ψ : X → F (Y ) is ψ : X → F (Y ) has a closed graph.
τ F -u.s.c. If Y is also locally compact, then every τ F -u.s.c.
/ ψ(x). Then, Proof. The first part of this proposition follows by [12, Proposition 2.1]. Let Y be locally compact, x ∈ X and y ∈ there exists an open set V such that K = V is compact and y ∈ V ⊂ K ⊂ Y \ ψ(x). Since ψ is τ F -u.s.c., U = X \ ψ −1 [ K ] is a neighbourhood of x. Thus, (x, y ) ∈ U × V and (U × V ) ∩ Graph(ψ) = ∅, so ψ has a closed graph. 2 We are going to prove the following slight generalisation of Theorem 1.4.
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Theorem 3.3. For a space X , the following are equivalent: (a) X is F -ultranormal. (b) If Y is completely metrizable and Φ : X → C (Y ) is l.s.c., then the mapping st[Φ, C ] has an l.s.c. closed-graph multi-selection ψ : X → C (C (Y ), τ V ). (c) If Y is completely metrizable, then every l.s.c. mapping Φ : X → C (Y ) has an l.s.c. closed-graph multi-section ψ : X → C (C (Y ), τ V ). (d) If Y is completely metrizable, then every l.s.c. mapping Φ : X → C (Y ) has a τ F -continuous multi-section ψ : X → C (C (Y ), τ V ). To prepare for the proof of Theorem 3.3, we proceed with some observations about point-finite covers. Proposition 3.4. For a space X , the following are equivalent: (a) For every point-finite open cover U of X , there exists a point-finite clopen cover { V W : W ∈ U F } of X such that V W ⊂ W , for all W ∈U F. (b) For every point-finite open cover U of X , there exists a point-finite clopen cover { V σ : σ ∈ [U ]<ω } of X such that V σ ⊂ σ , for < ω all σ ∈ [U ] . Proof. To show that (a) ⇒ (b), suppose that { V W : W ∈ U F } is apoint-finite clopen cover of X such that V W ⊂ W , for every W ∈ U F . Whenever W ∈ U F , let Γ W = {σ ∈ [U ]<ω : W = σ } and let σ ( W ) ∈ Γ W be a fixed element. Thus, we V σ = V W if σ = σ ( W ) and V σ = ∅ otherwise. get a partition {Γ W : W ∈ U F } of [U ]<ω . Finally, for every σ ∈ [U ]<ω , let Then, { V σ : σ ∈ [U ]<ω } is a point-finite clopen cover of X such that V σ ⊂ σ , for every σ ∈ [U ]<ω . Conversely, suppose that (b) holds and U is apoint-finite open cover of X . Then, U has an irreducible subcover O ⊂ U , see [9, Theorem 5.3.1]. σ1 = σ2 , whenever σ1 , σ2 ∈ [O ]<ω We now have that X = P for every proper subset P ⊂ O , and, in particular, F < ω are different elements. Hence, for every nonempty W ∈ O there exists a unique σ ( W ) ∈ [O ] , with W = σ ( W ). Thus, by (b), there exists a point-finite clopen cover { V W : W ∈ O F } of X such that V W ⊂ W for all W ∈ O F . Finally, set V W = ∅ for every W ∈ U F \ O F . Then, V W ⊂ W for all W ∈ U F . 2 Proposition 3.5. Let D be a set, Ω be a finite partition of D consisting of nonempty sets, and let A = {α ∈ [D ]<ω : every O ∈ Ω}. Then, [D ]<ω has a partition {π (α ): α ∈ A } such that π (α ) ⊂ [α ]<ω for every α ∈ A .
α ∩ O = ∅, for
] Proof. Let Π be the collection of all mapsp : B → 2[D such that B ⊂ A , the range { p (β): β ∈ B } of p is pairwise < ω <ω . For convenience, let dom( p ) = B be the domain of for all β ∈ B , and disjoint, p (β) ⊂ [β] β∈B p (β) = β∈B [β] p. Define a partial order in Π by letting for p , q ∈ Π that p q if dom( p ) ⊂ dom(q) and q is an extension of p, i.e. q(β) = p (β) for every β ∈ dom( p ). If Q ⊂ Π is a chain, then it defines a unique map p such that dom( p ) = q∈Q dom(q) and p dom(q) = q for every q ∈ Q . The properties by which Π was defined are preserved, hence p ∈ Π and we have that q p for every q ∈ Q . According to Zorn’s lemma, Π must have a maximal element π ∈ Π . The domain of π is A . Indeed, <ω by p dom(π ) = π and p (α ) = [α ]<ω \ β∈dom(π ) π (β). Thus, if α ∈ A \ dom(π ), then define p : dom(π ) ∪ {α } → 2[D ] of [D ]<ω is now the range of π . Namely, according to the p ∈ Π and π ≺ p which is impossible. partition The required <ω . If β ∈ [D ]<ω , then β is contained in some element of A π ∈ Π and, therefore, π ( α ) = [ α ] construction, α ∈A α ∈A and so β ∈ α ∈A [α ]<ω = α ∈A π (α ). 2 <ω
Proof of Theorem 3.3. (a) ⇒ (b). Let X be F -ultranormal, (Y , ρ ) be a complete metric space, and Φ : X → C (Y ) be an l.s.c. mapping. Also, let M : T → 2Y be a nonempty-open-valued locally-finite sieve on Y such that diamρ (M (t )) < 2−n , for every t ∈ T (n) and n < ω . Define a sieve R : Σ T → 2C (Y ) on (C (Y ), τ V ) by R (σ ) = M (σ ), for every σ ∈ Σ T . Next, define a sieve S : Σ T → 2 X on X by S (σ ) = (st[Φ, C ])−1 [R (σ )], σ ∈ Σ T . By Proposition 2.3, S is open-valued and
S (σ ) =
Φ −1 M (t ) : t ∈ σ ,
for every σ ∈ Σ T .
(3.1)
Since Φ is compact-valued and M is a locally-finite sieve, {Φ −1 [M (t )]: t ∈ T (0)} is a point-finite cover of X ; it is also open because Φ is l.s.c. Since X is F -ultranormal and Σ T (0) = [ T (0)]<ω , by (3.1) and Proposition 3.4, X has a point-finite clopen (σ ): σ ∈ Σ T (0)} such that L (σ ) ⊂ S (σ ), σ ∈ Σ T (0). Take an element σ ∈ Σ T (0), and set Ω = {node T (t ): t ∈ σ } cover {L and D = Ω . Then, Ω is a finite partition of D into nonempty sets, while, by virtue of (2.1),
A = α ∈ [D ]<ω : α ∩ O = ∅, for every O ∈ Ω
<ω node T (t ) : α ∩ node T (t ) = ∅, for every t ∈ σ = α∈ t ∈σ
= nodeΣT (σ ).
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By Proposition 3.5, [D ]<ω can be partitioned into sets π (μ), μ ∈ nodeΣT (σ ), such that π (μ) ⊂ [μ]<ω , for each μ ∈ nodeΣT (σ ). By construction, L (σ ) is clopen in X , hence it is F -ultranormal as well. Also, {Φ −1 [M (d)]: d ∈ D } is a pointfinite open cover of L (σ ) being a cover of S ( σ ). Therefore, by Proposition 3.4, there exists a point-finite clopen cover {Q (δ): δ ∈ [D ]<ω } of L (σ ) such that Q (δ) ⊂ d∈δ Φ −1 [M (d)], δ ∈ [D ]<ω . Finally, set L (μ) = δ∈π (μ) Q (δ), for each μ ∈ nodeΣT (σ ). Then, {L (μ): μ ∈ nodeΣT (σ )} is a point-finite clopen cover of L (σ ) because so is {Q (δ): δ ∈ [D ]<ω } and the sets π (μ), μ ∈ nodeΣT (σ ), form a partition of [D ]<ω . Take μ ∈ nodeΣT (σ ) and δ ∈ π (μ). Since π (μ) ⊂ [μ]<ω , by (3.1), we have that
Q (δ) ⊂
−1
Φ −1 M (d) ⊂ Φ M (t ) = S (μ). t ∈μ
d∈δ
Thus, L (μ) =
δ∈π (μ) Q (δ)
⊂ S (μ), and the construction of the sieve over the nodeΣT (σ ) is completed. Proceeding by induction on the levels of the tree Σ T , there exists a clopen-valued point-finite sieve L : Σ T → 2 X on X such that L (μ) ⊂ S (μ), for every μ ∈ Σ T . By Proposition 2.1, ΩR : B (ΣT ) → (C (Y ), τ V ) is a continuous perfect map. Then, define a mapping ψ : X → 2C (Y ) by
ψ(x) = ΩR 0L (x) = ΩR (β): β ∈ 0L (x) ,
x ∈ X.
According to [11, Propositions 5.1 and 5.2], 0L is nonempty-compact-valued and l.s.c., while, by [12, Corollary 4.6], it has a closed graph. Hence, ψ is l.s.c., and, by [12, Proposition 2.3], it has also a closed graph because ΩR is perfect. Since R is a (λ)-sieve on (C (Y ), τ V ), by [11, Lemma 7.1], ψ is a multi-selection for st[Φ, C ]. The implication (b) ⇒ (c) follows by Proposition 3.1, while (c) ⇒ (d) follows by Proposition 3.2. We complete the proof by showing that (d) ⇒ (a). Assume that X is as in (d), and U is a point-finite open cover of X . Endow U with the discrete topology, and define an l.s.c. mapping Φ : X → C (U ) by letting Φ(x) = {U ∈ U : x ∈ U }, x ∈ X . Since the Vietoris topology on C (U ) is discrete, (C (U ), τ V ) is actually the set [U ]<ω endowed with the discrete topology. Then, by (d), there exists a τ F -continuous mapping ψ : X → C ([U ]<ω ) such that σ ∩ Φ(x) = ∅, for every σ ∈ ψ(x) and x ∈ X . Since ψ is both l.s.c. and τ F -u.s.c., the family V σ = ψ −1 (σ ), σ ∈ [U ]<ω , is a point-finite clopen cover of X . Let
U σ = Φ −1 [σ ] =
Φ −1 (U ): U ∈ σ = σ ∈ U F,
σ ∈ [U ]<ω .
If x ∈ V σ , then σ ∈ ψ(x) and, therefore, σ ∩ Φ(x) = ∅. Hence, there is U ∈ σ , with U ∈ Φ(x), and we have that x ∈ Φ −1 (U ) ⊂ U σ . That is, V σ ⊂ U σ and, by Proposition 3.4, X is F -ultranormal. 2 By Theorem 3.3, we get the following consequence which furnishes the proof of Corollary 1.5. Corollary 3.6. For a space X , the following are equivalent: (a) X is F -ultrametacompact. (b) If Y is completely metrizable and Φ : X → F (Y ) is l.s.c., then the mapping st[Φ, C ] has an l.s.c. closed-graph multi-selection ψ : X → C (C (Y ), τ V ). (c) If Y is completely metrizable, then every l.s.c. mapping Φ : X → F (Y ) has an l.s.c. closed-graph multi-section ψ : X → C (C (Y ), τ V ). (d) If Y is completely metrizable, then every l.s.c. mapping Φ : X → F (Y ) has a τ F -continuous multi-section ψ : X → C (C ( Z ), τ V ). Proof. If X is a metacompact space, Y is a completely metrizable space and Φ : X → F (Y ) is l.s.c., by a result of Choban [6, Theorem 6.1], Φ has an l.s.c. multi-selection Ψ : X → C (Y ). Since st[Ψ, C ] is a multi-selection for st[Φ, C ], the implication (a) ⇒ (b) follows by Theorem 3.3. The implications (b) ⇒ (c) ⇒ (d) follow just like before by Propositions 3.1 and 3.2. As for (d) ⇒ (a), by Theorem 3.3, X is F -ultranormal. To show that X is metacompact, take an l.s.c. mapping Φ : X → F (Y ), where Y iscompletely metrizable. By (d), Φ has a τ F -continuous multi-section ψ : X → C (C ( Z ), τ V ). Define ϕ : X → C (Y ) by ϕ (x) = ψ(x), x ∈ X , see [17, Theorems 2.5]. Then, ϕ remains l.s.c., see [17, Theorems 5.7]. Thus, ϕ is an l.s.c. compactvalued section for Φ , and, by [12, Theorem 6.2], X is metacompact. 2 We conclude this section with an example showing that there are supermetacompact spaces which are not F ultrametacompact. Example 3.7. There exists a supermetacompact space X which is not F -ultrametacompact. Proof. We use virtually the same space X as given in [4, Example 4.2]. Namely, let X = [0, 1]2 \ {(0, 0)}, and for each x = 0, let H x = [0, 1] × {x} be the horizontal segment through the point (0, x), and V x = {x} × [0, 1] — the corresponding vertical one through the point (x, 0). Endow X with the topology in which all points (x, y ), x = 0 = y, are isolated, while the neighbourhoods of (0, x) and (x, 0), x = 0, consist of all but finitely many points of H x and, respectively, V x . Thus, if U ⊂ X
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is open and (0, x) ∈ U for some x = 0, then H x \ U is finite; similarly, V x \ U is finite if (x, 0) ∈ U . This implies that each open cover of X has a point-finite clopen refinement and, in particular, X is supermetacompact. However, X is not even F -ultranormal. Indeed, consider the point-finite open cover U of X , where
U = [0, 1] × (2−(n+1) , 2−n ], (2−(n+1) , 2−n ] × [0, 1]: n < ω .
Since U is countable, so is U F . We are only left to show that one cannot find a countable clopen refinement of U F . This follows from the observation that for any element U ∈ U F , any clopen subset W ⊂ U may contain only finitely many points of the type (0, x) or (x, 0) for x = 0. Consequently, one needs uncountably many such clopen subsets to cover the entire space X . 2 4. Sections, multi-sections and covering properties In this section, we discuss the relationship between sections and multi-sections of set-valued mappings, and their role in characterising covering properties. Our first result shows that, in the presence of τ V -continuity, sections and multi-sections are actually equivalent. Proposition 4.1. A mapping Φ : X → F (Y ) has a τ V -continuous multi-section ψ : X → C (C (Y ), τ V ) if and only if it has a τ V continuous section ϕ : X → C (Y ). In particular, a space X is superparacompact if and only if for every completely metrizable Y , every l.s.c. mapping Φ : X → F (Y ) has a τ V -continuous multi-section ψ : X → C (C (Y ), τ V ). Proof. If ϕ : X → C (Y ) is a τ V -continuous section for Φ , then ϕ : X → (C (Y ), τ V ) is continuous. In this case, the singletonvalued mapping ψ(x) = {ϕ (x)}, x ∈ X , is also τ V -continuous as a mapping from X to the compact subsets of (C (Y ), τ V ), and is a multi-section for Φ . Suppose that ψ : X → C (C (Y ), τ V ) is a τ V -continuous multi-section for Φ , and define a section ϕ : X → 2Y for Φ by ϕ (x) = ψ(x), x ∈ X . According to [17, Theorems 2.5 and 5.7], ϕ : X → C (Y ) and is τ V -continuous, which completes the first part of the proof. The second one now follows by Theorem 1.3. 2 Turning to the role of multi-sections in Theorem 3.3 and Corollary 3.6, let us explicitly mention that it is not clear if the simplistic approach used in Proposition 4.1 can be extended to τ F -continuous mappings. In fact, we have two different topologies, namely the Vietoris topology on C (Y ) and the Fell one on C (C (Y ), τ V ). In this regard, we have the following question. Question 1. Under what conditions on a space X , for every completely metrizable space Y , every l.s.c. mapping Φ : X → F (Y ) has a τ F -continuous section ϕ : X → C (Y )? We now turn to another interpretation of multi-sections. Proposition 4.2. For a metrizable space Y , a mapping Φ : X → F (Y ) has a τ F -continuous multi-section ψ : X → C (C (Y ), τ V ) if and only if there exists a metrizable space Z , a τ F -continuous mapping ϕ : X → C ( Z ) and a continuous perfect θ : Z → (C (Y ), τ V ) such that θ( z) ∩ Φ(x) = ∅ for every z ∈ ϕ (x) and x ∈ X . Proof. Let ψ : X → C (C (Y ), τ V ) be a τ F -continuous multi-section for Φ , Z be the space (C (Y ), τ V ), and θ : Z → C (Y ) be the identity map. Then, ϕ = ψ : X → C ( Z ) is τ F -continuous, while θ : Z → (C (Y ), τ V ) is a perfect continuous map (being a homeomorphism). Also, we have θ( z) ∩ Φ(x) = ∅ for every z ∈ ϕ (x) and x ∈ X . Conversely, let Z be a metrizable space, ϕ : X → C ( Z ) be τ F -continuous, and θ : Z → (C (Y ), τ V ) be a continuous perfect map such that θ(z) ∩ Φ(x) = ∅ for every z ∈ ϕ (x) and x ∈ X . Define ψ : X → C (C (Y ), τ V ) by ψ(x) = θ(ϕ (x)) = {θ( z): z ∈ ϕ (x)}, x ∈ X . Then, ψ remains l.s.c. as a composition of an l.s.c. mapping with a continuous one. Since θ is perfect, ψ is also τ F -u.s.c. If K ∈ ψ(x), then K = θ( z) for some z ∈ Z , and we have K ∩ Φ(x) = θ( z) ∩ Φ(x) = ∅. The proof is completed. 2 According to Corollary 1.5, we have the following immediate consequence. Corollary 4.3. A space X is F -ultrametacompact if and only if for every completely metrizable space Y and l.s.c. mapping Φ : X → F (Y ), there exists a metrizable space Z , a perfect continuous map θ : Z → (C (Y ), τ V ) and a τ F -continuous mapping ϕ : X → C ( Z ) such that θ(z) ∩ Φ(x) = ∅ for every z ∈ ϕ (x) and x ∈ X . This approach to multi-sections allows to extend some of the results for F -ultrametacompact spaces to supermetacompact spaces. Theorem 4.4. Let X be a supermetacompact space, Y be a completely metrizable space, and Φ : X → F (Y ) be an l.s.c. mapping. Then, there is a metrizable space Z , a τ F -continuous mapping ϕ : X → C ( Z ) and a τ V -continuous one θ : Z → C (Y ) such that θ(z) ∩ Φ(x) = ∅ for every z ∈ ϕ (x) and x ∈ X .
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Proof. Let ρ be a complete metric on Y , and M : T → 2Y be a nonempty-open-valued locally-finite sieve on Y with diamρ (M (t )) < 2−n , for every t ∈ T (n) and n < ω . Define a sieve R : Σ T → 2C (Y ) on (C (Y ), τ V ) by R (σ ) = M (σ ), for every σ ∈ Σ T . Next, define S : Σ T → 2 X by S (σ ) = (st[Φ, C ])−1 [R (σ )], σ ∈ Σ T . Then, by Proposition 2.3, S is an openvalued finitely additive sieve on X . We are going to construct a pruned tree ( D , ), a monotone map h : D → Σ T , and a point-finite clopen-valued sieve L : D → 2 X such that L is a multi-selection for S ◦ h. Here, h is monotone if h(s) ≺ h(t ) for every s, t ∈ D, with s ≺ t. Since {S (σ ): σ ∈ Σ T (0)} is a finitely additive open cover of X and X is supermetacompact, there exists a point-finite clopen cover { L d : d ∈ D 0 } of X and a map h0 : D 0 → Σ T (0) with L d ⊂ S (h0 (d)) for all d ∈ D 0 . We can proceed by induction. Namely, whenever p ∈ D 0 , consider the open cover {S (σ ) ∩ L p : σ ∈ nodeΣT (h0 ( p ))} of L p , which is finitely additive. Since L p is supermetacompact (being a clopen subset of X ), there exists a clopen point-finite cover { L d : d ∈ D ( p ,1) } of L p and a map h( p ,1) : D ( p ,1) → nodeΣT (h0 ( p )) such that L d ⊂ S (h( p ,1) (d)) ∩ L p for every d ∈ D ( p ,1) . Let D 1 be the disjoint union of the sets D ( p ,1) , p ∈ D 0 , and let h1 : D 1 → Σ T and π0 : D 1 → D 0 be defined by h1 D ( p ,1) = h( p ,1) , p ∈ D 0 , and respectively π0 (d) = p whenever d ∈ D ( p ,1) . This defines a point-finite clopen cover { L d : d ∈ D 1 } of X such
that L d ⊂ S (h1 (d)), d ∈ D 1 , and L p = { L d : d ∈ π0−1 ( p )}, p ∈ D 0 . Thus, by induction, for every n < ω we can define a set D n , maps hn : D n→ Σ T (n) and πn : D n+1 → D n , and a clopen point-finite cover { L d : d ∈ D n } of X such that L p ⊂ −1 S (hn ( p )) and L p = { L d : d ∈ πn ( p )}, p ∈ D n . Now, assuming that the sets D n , n < ω , are pairwise disjoint, we may define D = { D n : n < ω}, and a partial order “” on D such that ( D , ) is a tree for which D (n) = D n for each n < ω , and node D (d) = πn−1 (d) for each d ∈ D (n). Finally, define h : D → Σ T by h D n = hn , n < ω , and L : D → 2 X by L (d) = L d for d ∈ D. This completes the construction. Having already constructed these D, h and L , we conclude the proof by taking Z = B ( D ) and ϕ = 0L : X → C ( Z ). Then, ϕ is an l.s.c. closed-graph mapping (hence, τ F -continuous by Proposition 3.2), see the proof of (a) ⇒ (b) in Theorem 3.3. According to the construction above, h(β) ∈ B (Σ T ) for each β ∈ B ( D ). Moreover, the map h : B ( D ) → B (Σ T ) is continuous, see [11] and [14, Proposition 4.1]. By Proposition 2.1, the polar mapping ΩR : B (Σ T ) → C (Y ) is τ V -continuous, hence so is the composite mapping θ = ΩR ◦ h : Z → C (Y ). Since each
d∈β
R (h(d)) = θ(β), β ∈ B ( D ) = Z , is
a singleton of C (Y ) and L (d) ⊂ S (h(d)) = (st[Φ, C ])−1 [R (h(d))], d ∈ D, by [11, Lemma 7.1], the composite mapping θ ◦ ϕ : X → C (C (Y ), τ V ) is a multi-selection for st[Φ, C ]. Here, (θ ◦ ϕ )(x) = θ(ϕ (x)) ∈ C (C (Y ), τ V ), x ∈ X . This actually completes the proof. Indeed, if x ∈ X , then θ(ϕ (x)) ⊂ st[Φ, C ](x) and, therefore, θ( z) ∩ Φ(x) = ∅ for every z ∈ ϕ (x). 2 As the reader may observe, the difference between Corollary 4.3 and Theorem 4.4 is that in Corollary 4.3 the mapping
θ : Z → (C (Y ), τ V ) is also perfect. However, the authors were unable to show if the converse of Theorem 4.4 holds. We have only the following result for the special case of a locally compact range. Theorem 4.5. A space X is supermetacompact if and only if for every l.s.c. mapping Φ : X → F (Y ), where Y is a locally compact paracompact space, there exists a discrete space Z , a τ F -continuous mapping ϕ : X → C ( Z ) and a mapping θ : Z → C (Y ) such that θ(z) ∩ Φ(x) = ∅ for every z ∈ ϕ (x) and x ∈ X . Proof. Suppose that X is supermetacompact, Y is a locally compact paracompact space, and Φ : X → F (Y ) is l.s.c. According to a result of Choban and Michael [7], there exists an ultraparacompact space T , a perfect continuous surjection g : T → Y and a subset S ⊂ T such that g ( S ) = Y and g S is open. This implies that T is also locally compact. Hence, it has a discrete partition K consisting of nonempty compact subsets. Let Z = K F ⊂ C ( T ) be equipped with the Vietoris topology. Since {γ : γ ∈ [K ]<ω } is an open partition of Z consisting of singletons, Z is discrete. Define θ : Z → C (Y ) by θ( z) = g ( z), z ∈ Z . Since Φ is l.s.c., U = {Φ −1 [ g ( K ∩ S )]: K ∈ K } is an open cover of X . Since X is supermetaU F has a point-finite clopen refinement V . For every V ∈ V , fix a nonempty finite subset γ ( V ) ⊂ K with compact, − V ⊂ {Φ 1 [ g ( K )]: K ∈ γ ( V )}, and let z V = γ ( V ). Thus, z V ∈ Z and V ⊂ Φ −1 [ g ( z V )] = Φ −1 [θ( z V )]. Finally, define a mapping ϕ : X → 2 Z by ϕ (x) = { z V : x ∈ V }, x ∈ X . Since V is open and point-finite, ϕ is l.s.c. and finite-valued (hence, compact-valued as well). Since V is also closed and Z is discrete, ϕ is τ F -u.s.c. These Z , ϕ and g are as required. Indeed, if x ∈ X and z V ∈ ϕ (x), then x ∈ V ⊂ Φ −1 [θ( z V )], and therefore θ( z V ) ∩ Φ(x) = ∅. Conversely, take an open cover U of X , and endow it with the discrete topology. Next, define an l.s.c. mapping Φ : X → F (U ) by Φ(x) = {U ∈ U : x ∈ U }, x ∈ X . Since U is locally compact, there exists a discrete space Z , a τ F continuous mapping ϕ : X → C ( Z ) and a mapping θ : Z → C (U ) such that θ( z) ∩ Φ(x) = ∅, for every z ∈ ϕ (x) and x ∈ X . Then, V = {ϕ −1 ( z): z ∈ Z } is a clopen point-finite refinement of U F . Indeed, if z ∈ Z and x ∈ ϕ −1 ( z), then z ∈ ϕ (x) and, therefore, x ∈ Φ −1 [θ( z)] because θ( z) ∩ Φ(x) = ∅. That is,
ϕ −1 (z) ⊂ Φ −1 θ(z) =
θ(z) ∈ U F .
2
According to [3, Proposition 3.2], a Tychonoff space X is supermetacompact if and only if for every compact B ⊂ β X \ X , ˇ there exists a point-finite clopen cover W of X such that B ∩ W β X = ∅ for every W ∈ W . Here, β X is the Cech–Stone β X is the closure of W in β X . Theorem 4.5 can be regarded as a natural extension of this compactification of X , while W characterisation. To this end, let F∅ (Y ) = F (Y ) ∪ {∅} and C∅ (Y ) = C (Y ) ∪ {∅}. The Fell topology was considered originally on F∅ (Y ), and it was proved in [10] that (F∅ (Y ), τ F ) is always compact regardless of what is Y , [10, Lemma 1]. It was
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further proved in [10, Theorem 1] that (F∅ (Y ), τ F ) is compact and Hausdorff provided that Y is locally compact. In this result, Y is not assumed even to be T 1 . Now, we have the following result. Corollary 4.6. A Tychonoff space X is supermetacompact if and only if for every compact set B ⊂ β X \ X there exists a discrete space Z and a Fell-continuous mapping ψ : β X → F∅ ( Z ) such that ψ( X ) ⊂ C ( Z ) and ψ( B ) ⊂ {∅}. Proof. Suppose that X is a supermetacompact Tychonoff space and B ⊂ β X \ X is compact. For every point x ∈ X take a neighbourhood U (x) with U (x)β X ∩ B = ∅. Set U = {U (x): x ∈ X }, and endow it with the discrete topology. Next, define an l.s.c. mapping Φ : X → F (U ) by Φ(x) = {U ∈ U : x ∈ U }, x ∈ X . Then, by Theorem 4.5, there exists a discrete space Z , a τ F -continuous mapping ϕ : X → C ( Z ) and a mapping θ : Z → C (U ) such that θ( z) ∩ Φ(x) = ∅ for every z ∈ ϕ (x). Next, define ϕ∅ : X ∪ B → C∅ ( Z ) by ϕ∅ X = ϕ and ϕ∅ (x) = ∅ for x ∈ B. The basic neighbourhoods of ∅ in C∅ ( Z ) are given by the collection { Z \ K } = { S ∈ C∅ ( Z ): S ∩ K = ∅}, where K runs over the compact subsets of Z . Take a compact (i.e., finite)
subset K ⊂ Z . Then, ϕ −1 [ K ] is a clopen subset of X , and we have that ϕ −1 [ K ]β X ∩ B = ∅ because ϕ −1 [ K ] is contained in finitely many elements of U , see the proof of Theorem 4.5. This implies that ϕ∅ is τ F -continuous. Since (F∅ ( Z ), τ F ) is a compact Hausdorff space (as mentioned above), ϕ∅ can be extended to a τ F -continuous mapping ψ : β X → F∅ ( Z ). We have ψ( B ) = ϕ∅ ( B ) ⊂ {∅} and ψ( X ) = ϕ∅ ( X ) ⊂ C ( Z ), so these Z and ψ are as required. To show the converse, let B ⊂ β X \ X be compact and let ψ : β X → F∅ ( Z ) be τ F -continuous, where Z is a discrete space, ψ( X ) ⊂ C ( Z ) and ψ( B ) ⊂ {∅}. Then, each set ψ −1 ( z), z ∈ Z , is clopen in β X because { y ∈ β X: z ∈ / ψ( y )} is a neighbourhood of ∅ in (F∅ ( Z ), τ F ). Thus, {ψ −1 (z) ∩ X: z ∈ Z } is a point-finite, clopen cover of X because ψ( X ) ⊂ C ( Z ). Finally, since ψ( B ) ⊂ {∅} ⊂ F∅ ( Z ) \ C ( Z ), for every z ∈ Z we have that
ψ −1 ( z ) ∩ X β X ⊂ ψ −1 ( z ) ⊂ β X \ B . According to [3, Proposition 3.2], X is supermetacompact.
2
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Topology and its Applications www.elsevier.com/locate/topol
Fell continuous multi-sections David Buhagiar ∗ , Valentin Gutev Department of Mathematics, Faculty of Science, University of Malta, Msida, MSD 2080, Malta
a r t i c l e
i n f o
Article history: Received 19 October 2012 Received in revised form 5 March 2013 Accepted 6 March 2013 MSC: 54C60 54C65 54B20 54D15
a b s t r a c t The paper is devoted to several characterisations of metacompact-like properties in terms of special sections continuous with respect to the Fell hyperspace topology. These results fit naturally within the framework of covering properties described by similar sections, but continuous with respect to other hyperspace topologies. © 2013 Elsevier B.V. All rights reserved.
Keywords: Set-valued mapping Hyperspace Vietoris topology Fell topology Lower semi-continuous Closed-graph Section F -ultranormal space
1. Introduction All spaces in this paper are assumed to be T 1 -spaces. For space Y , let 2Y be the power set of Y ; F (Y ) be the set of all nonempty closed subsets of Y ; and C (Y ) — that of all compact members of F (Y ). Recall that the Vietoris topology τ V on F (Y ) is generated by all collections of the form
V = S ∈ F (Y ): S ⊂ V and S ∩ V = ∅, whenever V ∈ V , where V runs over the finite families of open subsets of Y . Another topology on F (Y ) which will play an important role in this paper is the Fell topology τ F , it is generated by all basic Vietoris neighbourhoods V such that V is a finite family of open subsets of Y and Y \ V is compact. In what follows, let us make the explicit agreement that when the clarity seems to demand it, for a given topology τ on F (Y ) and Q ⊂ F (Y ), we will broadly use (Q , τ ) to express that Q is endowed with τ . For instance, we will often write ϕ : X → (C (Y ), τ ) for a map from X to the space (C (Y ), τ ); F (C (Y ), τ ) will mean the nonempty closed subsets of (C (Y ), τ ); etc. In particular, we shall say that a set-valued mapping ϕ : X → F (Y ) is τ -continuous if it is continuous as a usual map from X to the space (F (Y ), τ ). Here are some natural examples of τ -continuous mappings. A mapping
*
Corresponding author. E-mail addresses: [email protected] (D. Buhagiar), [email protected] (V. Gutev).
0166-8641/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.topol.2013.03.001
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ϕ : X → 2Y is lower semi-continuous, or l.s.c., if the set ϕ −1 [U ] = {x ∈ X: ϕ (x) ∩ U = ∅} is open in X for every open U ⊂ Y ; and ϕ is upper semi-continuous, or u.s.c., if ϕ −1 [ F ] is closed in X for every closed F ⊂ Y . Similarly, we will say that ϕ : X → F (Y ) is τ F -upper semi-continuous, or τ F -u.s.c., if ϕ −1 [ K ] is closed for every compact K ⊂ Y . We now have that ϕ : X → F (Y ) is τ V -continuous if and only if it is both l.s.c. and u.s.c.; and that ϕ is τ F -continuous if and only if it is both l.s.c. and τ F -u.s.c. There is a natural relationship between covering properties of topological spaces and selection type properties of setvalued mappings. Turning to this, let us recall that a map f : X → Y is a selection (or, a single-valued selection) for Φ : X → 2Y if f (x) ∈ Φ(x) for every x ∈ X . A set-valued mapping ϕ : X → 2Y is a multi-selection (or, a set-valued selection) for Φ : X → 2Y if ϕ (x) ⊂ Φ(x) for every x ∈ X ; and ϕ : X → 2Y is a section for Φ : X → 2Y if ϕ (x) ∩ Φ(x) = ∅ for every x ∈ X . If ϕ is a section for Φ , then both ϕ and Φ must be nonempty-valued. Of course, every nonempty-valued multi-selection for Φ is also a section for Φ . A space X is ultranormal if every two disjoint closed subsets are contained in disjoint clopen subsets, i.e. if the large inductive dimension of X is 0. In the realm of normal spaces, X is ultranormal if and only if its covering dimension is 0. The ultranormal paracompact spaces are often called ultraparacompact. As is well known, a Hausdorff space X is ultraparacompact if and only if every open cover of X has a pairwise disjoint open refinement, see [8]. The existence of such refinements is behind the following result. Theorem 1.1. A space X is ultraparacompact if and only if for every completely metrizable space Y , every l.s.c. mapping Φ : X → F (Y ) has a continuous selection (in particular, a τ V -continuous multi-selection). Theorem 1.1 is due to E. Michael [19, Theorem 2] (see also [18]) and T. Ishii [15, Theorem 3]. Using a classical result of Lefshetz [16] (see also [9, Theorem 1.5.18]), it follows that a space X is ultranormal if and only if for every point-finite open cover U of X there exists a clopen (point-finite) cover { V U : U ∈ U } of X such that V U ⊂ U , for all U ∈ U . In particular, X is an ultranormal metacompact space if and only if for every open cover U of X there exists a clopen point-finite cover { V U : U ∈ U } of X such that V U ⊂ U , for all U ∈ U . In view of this covering property, it makes sense to say that a space X is ultrametacompact if it is an ultranormal metacompact space. In slightly different terms (see, Proposition 3.2), the following result was obtained in [12, Theorem 5.2], see also [6, Theorem 6.1] and [12, Theorem 6.3]. Theorem 1.2. A space X is ultranormal if and only if for every metrizable space Y , every l.s.c. mapping Φ : X → C (Y ) has a τ F continuous multi-selection ϕ : X → C (Y ). In particular, X is ultrametacompact if and only if for every completely metrizable space Y , every l.s.c. mapping Φ : X → F (Y ) has a τ F -continuous multi-selection ϕ : X → C (Y ). Let W be a collection of subsets of a set X . If U , V ∈ W , then a finite sequence W 1 , W 2 , . . . , W k of elements of W is called a chain from U to V if U = W 1 , V = W k and W i ∩ W i +1 = ∅ for every i = 1, . . . , k − 1. A subset P ⊂ W is called connected if every pair of elements of P is connected by a chain. The components of W are defined as the maximal connected subsets of W . A space X is called superparacompact (Pasynkov, see [21]) if every open cover of X has an open finite component (i.e., having finite components) refinement. The following theorem was proved in [1]. Theorem 1.3. A space X is superparacompact if and only if for every completely metrizable space Y , every l.s.c. mapping Φ : X → F (Y ) has a τ V -continuous section ϕ : X → C (Y ).
W ∈ U for every finite W ⊂ U . For a collection U , let U F be the collection A cover U of X is finitely additive if of the unions of all finite subcollections of U . In these terms, a cover U of X is finitely additive if and only if U = U F . According to [2, Proposition 2.3] (see also [3, Theorem 2.2]), a space X is superparacompact if and only if every finitely additive cover U of X has a pairwise disjoint open refinement. In particular, a space X is superparacompact if and only if for every open cover U of X , the cover U F has a disjoint open refinement. Hence, superparacompact spaces can be regarded as “F -ultraparacompact”. This also reveals a natural relationship between Theorems 1.1 and 1.3, and the role of finitely additive covers to transform multi-selections into compact-valued sections. Motivated by this, we will say that a space X is F -ultranormal if for every point-finite open cover U of X , there exists a point-finite clopen cover { V W : W ∈ U F } of X with V W ⊂ W , for all W ∈ U F . The following finitely additive version of Theorem 1.2 will be proved in this paper. Theorem 1.4. A space X is F -ultranormal if and only if for every completely metrizable space Y , every l.s.c. mapping Φ : X → C (Y ) admits a τ F -continuous mapping ψ : X → C (C (Y ), τ V ) such that K ∩ Φ(x) = ∅ for every K ∈ ψ(x) and x ∈ X . Let us explicitly remark that “C (C (Y ), τ V )” is the set of all nonempty compact subsets of the space (C (Y ), τ V ). That is, ψ is a map from X to the Fell hyperspace of the Vietoris hyperspace. Motivated by Theorem 1.4, we shall say that a mapping ψ : X → 2C (Y ) is a multi-section for Φ : X → 2Y if K ∩ Φ(x) = ∅ for every K ∈ ψ(x) and x ∈ X . The interested reader is referred to Section 4, where we discuss the subtle difference between sections and multi-sections. Just like before, we will say that a space X is F -ultrametacompact if it is an F -ultranormal metacompact space. The following consequence now follows from Theorem 1.4.
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Corollary 1.5. A space X is F -ultrametacompact if and only if for every completely metrizable space Y , every l.s.c. mapping Φ : X → F (Y ) admits a τ F -continuous multi-section ψ : X → C (C (Y ), τ V ). A space X is called supermetacompact [3, Definition 3.1] if every finitely additive open cover of X has a point-finite clopen refinement. Every ultrametacompact space is F -ultrametacompact, and every F -ultrametacompact space is supermetacompact, but neither of these is invertible. Indeed, every compact space is F -ultrametacompact, but there are compact spaces which are not ultranormal, for instance the closed unit interval. According to Example 3.7, there exists a supermetacompact space which is not F -ultrametacompact. Concerning multi-sections, we were able to provide a similar characterisation of supermetacompactness when the set-valued mappings take values in locally compact spaces, see Theorem 4.5. In general, for a completely metrizable range, we have only a partial result for these spaces, see Theorem 4.4. The proof of Theorem 1.4 is finalised in Section 3 (see, Theorem 3.3), while Corollary 1.5 is a partial case of a slightly more general result (see, Corollary 3.6). The proof of Theorem 1.4 is reduced to the existence of multi-selections for special star-mappings associated to set-valued mappings (Proposition 3.1). These multi-selections are constructed using special additive sieves which are developed in the next section. Another important ingredient in this proof is played by closedgraph mappings which are used as an interface to τ F -continuity, see Proposition 3.2. 2. Special additive sieves First of all, we need some standard terminology for set-valued mappings. Given Φ : X → 2Y , the inverse Φ −1 : Y → 2 X of Φ isdefined by Φ −1 ( y ) = Φ −1 [{ y }], y ∈ Y . In particular, we always have that (Φ −1 )−1 = Φ . Whenever S ⊂ X , let Φ[ S ] = {Φ(x): x ∈ S } which is a subset of Y and will play the role of the image of S by Φ , when Φ is considered as a set-valued mapping. Let us explicitly remark that we will also use Φ( S ) = {Φ(x): x ∈ S } which is a subset of 2Y , in fact it is the image of S by Φ , when Φ is considered as a usual map from X to the set 2Y . To every mapping Φ : X → 2Y we will associate another one Φ : X → 2Y defined by Φ(x) = Φ(x), x ∈ X . The mapping Φ will play the role of a point-wise closure of Φ . We now turn to some basic terminology for sieves. A partially ordered set ( T , ) is a tree if {s ∈ T : s t } is well-ordered for every t ∈ T . In this case, we use T (0) to denote the set of the minimal elements of T . Also, given an ordinal α , if T (β) is defined for every β < α , then T (α ) denotes the minimal elements of T \ ( T α ), where T α = { T (β): β < α }. The set T (α ) is called the α th-level of T , while the height of T is the least ordinal α such that T α = T . A maximal linearly ordered subset of T is called a branch, and B ( T ) is used to denote the set of all branches of T . Finally, for t ∈ T , the node of t in T is the subset node T (t ) ⊂ T of all immediate successors of t; for convenience, we also set node T (∅) = T (0). A tree ( T , ) is pruned if every element of T has a successor in T , i.e. node T (t ) = ∅ for every t ∈ T . Given a set Y and a pruned tree ( T , ) of height ω , a mapping S : T → 2Y is a sieve on Y if Y = S [node T (∅)] and S (t ) = S [nodeT (t )] forevery t ∈ T . For a tree ( T , ) and S : T → 2Y , the polar mapping ΩS : B ( T ) → 2Y associated to S −1 is defined by ΩS (β) = {S (t ): t ∈ β}, β ∈ B ( T ); and the inverse polar mapping 0S : Y → 2B( T ) is defined by 0S = ΩS . Following [5], a nonempty-open-valued sieve S : T → 2Y on a space Y is called a (λ)-sieve if for every β ∈ B ( T ) there exists an y (β) ∈ Y such that if U is a neighbourhood of y (β), then S (t ) ⊂ U for some t ∈ β . This clearly implies that y (β) ∈ S (t ) = S (t ) for every t ∈ β . Hence, for a Hausdorff space Y , we have that ΩS (β) = { y (β)} and, therefore, the polar mapping ΩS : B ( T ) → 2Y is singleton-valued. Since every singleton-valued mapping has the same graph as a singlevalued one (representing the same relation), we will make no difference between such mappings. Thus, in this case, the polar mapping ΩS : B ( T ) → Y is a usual map. A sieve S : T → 2Y on a space Y is locally-finite (point-finite, etc.) if each cover {S (t ): t ∈ T (n)}, n < ω , is locally-finite (point-finite, etc.). Let us explicitly mention the well-known fact that every complete metric space (Y , ρ ) has a nonemptyopen-valued locally-finite (λ)-sieve. For instance, use [20, Lemma 2.2] to get a nonempty-open-valued locally-finite sieve S : T → 2Y such that diamρ (S (t )) < 2−n for every t ∈ T (n) and n < ω . Then, S is a (λ)-sieve by Cantor’s intersection theorem. Following Nyikos [22], for a tree ( T , ) and t ∈ T , let
O (t ) = β ∈ B ( T ): t ∈ β .
If T is pruned and of height ω , the family {O (t ): t ∈ T } is a base for a completely metrizable (non-Archimedean) topology on B ( T ). We will refer to this topology as the branch topology, and to the resulting topological space as the branch space. Throughout this paper, B ( T ) will be always endowed with the branch topology when it comes to consider it as a topological space. It is well known that B ( T ) is compact if and only if all levels of T are finite. For a set D and a cardinal number κ ω , let [ D ]<κ = { S ⊂ D: 1 | S | < κ }. Following [13], for a pruned tree ( T , ) of height ω , we consider another tree (Σ T , ) defined in the following way. Let Σ T = {[ T (n)]<ω : n < ω}. Next, define a relation on Σ T by letting for σ , μ ∈ Σ T that σ ≺ μ if
μ⊂
node T (s): s ∈ σ
and
μ ∩ nodeT (s) = ∅, s ∈ σ .
(2.1)
Finally, extend this relation to a partial order on Σ T making it transitive. Thus, we get a pruned tree (Σ T , ) of height because so is ( T , ).
ω
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Proposition 2.1. Let (Y , ρ ) be a complete metric space, and M : T → 2Y be a nonempty-open-valued locally-finite sieve on Y such that diamρ (M (t )) < 2−n , for every t ∈ T (n) and n < ω . Define a mapping R : Σ T → 2C (Y ) by letting R (σ ) = M (σ ), for every σ ∈ ΣT . Then, R is an open-valued locally-finite (λ)-sieve on (C (Y ), τ V ). In particular, ΩR : B (ΣT ) → (C (Y ), τ V ) is a continuous perfect map. Proof. The fact that R is a sieve on C (Y ) follows by (2.1). According to the definition of R , we also have that it is openvalued and locally-finite. It is well known that the Vietoris topology on C (Y ) is metrizable with respect to the Hausdorff metric H (ρ ) on C (Y ) associated to ρ :
H (ρ )( S , T ) = max
ρ ( S , y ) + ρ ( y , T ): y ∈ S ∪ T ,
S , T ∈ C (Y ).
Since ρ is complete, so is the Hausdorff metric H (ρ ) on C (Y ). If σ ∈ Σ T , then σ belongs to the nth-level of the tree Σ T for some n < ω . We now have that H (ρ )( K 1 , K 2 ) < 2−n for every K 1 , K 2 ∈ R (σ ) = {M (t ): t ∈ σ } because diamρ (M (t )) < 2−n for all t ∈ σ . That is, diam H (ρ ) (R (σ )) 2−n and, by Cantor’s intersection theorem, R is a (λ)-sieve on C (Y ). Hence,
ΩR : B (Σ T ) → (C (Y ), τ V ) is a continuous map, see [11, Proposition 5.2]. Since R is a locally-finite closed-valued sieve 2
on C (Y ), the inverse 0R = Ω −1 is compact-valued and u.s.c. [11, Lemma 5.3], therefore ΩR is also perfect. R
To every mapping Φ : X → F (Y ) we are going to associate another one st[Φ, C ] : X → 2C (Y ) by letting for x ∈ X that
st[Φ, C ](x) = K ∈ C (Y ): K ∩ Φ(x) = ∅ .
(2.2)
The mapping st[Φ, C ] will play the role of the star of Φ with respect to the compact subsets of Y . The following propositions collect some basic properties of st[Φ, C ]; the verification of the first one is left to the reader. Proposition 2.2. If Φ : X → F (Y ) and C (Y ) is endowed with the Vietoris topology, then st[Φ, C ] : X → F (C (Y ), τ V ) and it is l.s.c. if and only if Φ is l.s.c. Proposition 2.3. Let Y be a metrizable space, M : T → 2Y be a nonempty-open-valued locally-finite sieve on Y , and let R : Σ T → 2C (Y ) be the sieve on (C (Y ), τ V ) defined by R (σ ) = M (σ ), for every σ ∈ Σ T . Also, let Φ : X → F (Y ) be an l.s.c. mapping. Define S : Σ T → 2 X by S (σ ) = (st[Φ, C ])−1 [R (σ )], σ ∈ Σ T . Then, S is an open-valued sieve on X such that
S (σ ) =
Φ −1 M (t ) : t ∈ σ ,
for every σ ∈ Σ T .
Proof. According to Proposition 2.2, S is an open-valued sieve on X . If σ ∈ Σ T and st[Φ, C ](x) ∩ R (σ ) = ∅, then there exists K ∈ C (Y ) such that K ∩ Φ(x) = ∅ and K ∈ {M (t ): t ∈ σ }, hence x ∈ {Φ −1 [M (t )]: t ∈ σ }. Conversely, if Φ(x) ∩ M (t ) = ∅ for some t ∈ σ , then there is a finite set F ⊂ Y such that F ∈ {M (t ): t ∈ σ } and F ∩ Φ(x) = ∅, i.e. x ∈ S (σ ) = (st[Φ, C ])−1 [R (σ )]. Thus, S (σ ) = {Φ −1 [M (t )]: t ∈ σ }, σ ∈ Σ T . 2 3. Metacompact-like spaces and multi-sections Recall that a mapping ψ : X → 2C (Y ) is a multi-section for Φ : X → 2Y if K ∩ Φ(x) = ∅, for every K ∈ ψ(x) and x ∈ X . Multi-sections are in good accordance with the star-mapping st[Φ, C ] defined in the previous section, see (2.2). Namely, the following observation follows immediately from the definitions. Proposition 3.1. For mappings Φ : X → F (Y ) and ψ : X → C (C (Y ), τ V ), the following are equivalent: (a) ψ is a multi-section for Φ . (b) ψ is a multi-selection for st[Φ, C ]. A mapping ψ : X → 2Y has a closed graph, or is a closed-graph mapping, if its graph Graph(ψ) = {(x, y ) ∈ X × Y : y ∈ ψ(x)}, is closed in X × Y . Proposition 3.2. Every closed-graph mapping ψ : X → F (Y ) is ψ : X → F (Y ) has a closed graph.
τ F -u.s.c. If Y is also locally compact, then every τ F -u.s.c.
/ ψ(x). Then, Proof. The first part of this proposition follows by [12, Proposition 2.1]. Let Y be locally compact, x ∈ X and y ∈ there exists an open set V such that K = V is compact and y ∈ V ⊂ K ⊂ Y \ ψ(x). Since ψ is τ F -u.s.c., U = X \ ψ −1 [ K ] is a neighbourhood of x. Thus, (x, y ) ∈ U × V and (U × V ) ∩ Graph(ψ) = ∅, so ψ has a closed graph. 2 We are going to prove the following slight generalisation of Theorem 1.4.
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Theorem 3.3. For a space X , the following are equivalent: (a) X is F -ultranormal. (b) If Y is completely metrizable and Φ : X → C (Y ) is l.s.c., then the mapping st[Φ, C ] has an l.s.c. closed-graph multi-selection ψ : X → C (C (Y ), τ V ). (c) If Y is completely metrizable, then every l.s.c. mapping Φ : X → C (Y ) has an l.s.c. closed-graph multi-section ψ : X → C (C (Y ), τ V ). (d) If Y is completely metrizable, then every l.s.c. mapping Φ : X → C (Y ) has a τ F -continuous multi-section ψ : X → C (C (Y ), τ V ). To prepare for the proof of Theorem 3.3, we proceed with some observations about point-finite covers. Proposition 3.4. For a space X , the following are equivalent: (a) For every point-finite open cover U of X , there exists a point-finite clopen cover { V W : W ∈ U F } of X such that V W ⊂ W , for all W ∈U F. (b) For every point-finite open cover U of X , there exists a point-finite clopen cover { V σ : σ ∈ [U ]<ω } of X such that V σ ⊂ σ , for < ω all σ ∈ [U ] . Proof. To show that (a) ⇒ (b), suppose that { V W : W ∈ U F } is apoint-finite clopen cover of X such that V W ⊂ W , for every W ∈ U F . Whenever W ∈ U F , let Γ W = {σ ∈ [U ]<ω : W = σ } and let σ ( W ) ∈ Γ W be a fixed element. Thus, we V σ = V W if σ = σ ( W ) and V σ = ∅ otherwise. get a partition {Γ W : W ∈ U F } of [U ]<ω . Finally, for every σ ∈ [U ]<ω , let Then, { V σ : σ ∈ [U ]<ω } is a point-finite clopen cover of X such that V σ ⊂ σ , for every σ ∈ [U ]<ω . Conversely, suppose that (b) holds and U is apoint-finite open cover of X . Then, U has an irreducible subcover O ⊂ U , see [9, Theorem 5.3.1]. σ1 = σ2 , whenever σ1 , σ2 ∈ [O ]<ω We now have that X = P for every proper subset P ⊂ O , and, in particular, F < ω are different elements. Hence, for every nonempty W ∈ O there exists a unique σ ( W ) ∈ [O ] , with W = σ ( W ). Thus, by (b), there exists a point-finite clopen cover { V W : W ∈ O F } of X such that V W ⊂ W for all W ∈ O F . Finally, set V W = ∅ for every W ∈ U F \ O F . Then, V W ⊂ W for all W ∈ U F . 2 Proposition 3.5. Let D be a set, Ω be a finite partition of D consisting of nonempty sets, and let A = {α ∈ [D ]<ω : every O ∈ Ω}. Then, [D ]<ω has a partition {π (α ): α ∈ A } such that π (α ) ⊂ [α ]<ω for every α ∈ A .
α ∩ O = ∅, for
] Proof. Let Π be the collection of all mapsp : B → 2[D such that B ⊂ A , the range { p (β): β ∈ B } of p is pairwise < ω <ω . For convenience, let dom( p ) = B be the domain of for all β ∈ B , and disjoint, p (β) ⊂ [β] β∈B p (β) = β∈B [β] p. Define a partial order in Π by letting for p , q ∈ Π that p q if dom( p ) ⊂ dom(q) and q is an extension of p, i.e. q(β) = p (β) for every β ∈ dom( p ). If Q ⊂ Π is a chain, then it defines a unique map p such that dom( p ) = q∈Q dom(q) and p dom(q) = q for every q ∈ Q . The properties by which Π was defined are preserved, hence p ∈ Π and we have that q p for every q ∈ Q . According to Zorn’s lemma, Π must have a maximal element π ∈ Π . The domain of π is A . Indeed, <ω by p dom(π ) = π and p (α ) = [α ]<ω \ β∈dom(π ) π (β). Thus, if α ∈ A \ dom(π ), then define p : dom(π ) ∪ {α } → 2[D ] of [D ]<ω is now the range of π . Namely, according to the p ∈ Π and π ≺ p which is impossible. partition The required <ω . If β ∈ [D ]<ω , then β is contained in some element of A π ∈ Π and, therefore, π ( α ) = [ α ] construction, α ∈A α ∈A and so β ∈ α ∈A [α ]<ω = α ∈A π (α ). 2 <ω
Proof of Theorem 3.3. (a) ⇒ (b). Let X be F -ultranormal, (Y , ρ ) be a complete metric space, and Φ : X → C (Y ) be an l.s.c. mapping. Also, let M : T → 2Y be a nonempty-open-valued locally-finite sieve on Y such that diamρ (M (t )) < 2−n , for every t ∈ T (n) and n < ω . Define a sieve R : Σ T → 2C (Y ) on (C (Y ), τ V ) by R (σ ) = M (σ ), for every σ ∈ Σ T . Next, define a sieve S : Σ T → 2 X on X by S (σ ) = (st[Φ, C ])−1 [R (σ )], σ ∈ Σ T . By Proposition 2.3, S is open-valued and
S (σ ) =
Φ −1 M (t ) : t ∈ σ ,
for every σ ∈ Σ T .
(3.1)
Since Φ is compact-valued and M is a locally-finite sieve, {Φ −1 [M (t )]: t ∈ T (0)} is a point-finite cover of X ; it is also open because Φ is l.s.c. Since X is F -ultranormal and Σ T (0) = [ T (0)]<ω , by (3.1) and Proposition 3.4, X has a point-finite clopen (σ ): σ ∈ Σ T (0)} such that L (σ ) ⊂ S (σ ), σ ∈ Σ T (0). Take an element σ ∈ Σ T (0), and set Ω = {node T (t ): t ∈ σ } cover {L and D = Ω . Then, Ω is a finite partition of D into nonempty sets, while, by virtue of (2.1),
A = α ∈ [D ]<ω : α ∩ O = ∅, for every O ∈ Ω
<ω node T (t ) : α ∩ node T (t ) = ∅, for every t ∈ σ = α∈ t ∈σ
= nodeΣT (σ ).
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By Proposition 3.5, [D ]<ω can be partitioned into sets π (μ), μ ∈ nodeΣT (σ ), such that π (μ) ⊂ [μ]<ω , for each μ ∈ nodeΣT (σ ). By construction, L (σ ) is clopen in X , hence it is F -ultranormal as well. Also, {Φ −1 [M (d)]: d ∈ D } is a pointfinite open cover of L (σ ) being a cover of S ( σ ). Therefore, by Proposition 3.4, there exists a point-finite clopen cover {Q (δ): δ ∈ [D ]<ω } of L (σ ) such that Q (δ) ⊂ d∈δ Φ −1 [M (d)], δ ∈ [D ]<ω . Finally, set L (μ) = δ∈π (μ) Q (δ), for each μ ∈ nodeΣT (σ ). Then, {L (μ): μ ∈ nodeΣT (σ )} is a point-finite clopen cover of L (σ ) because so is {Q (δ): δ ∈ [D ]<ω } and the sets π (μ), μ ∈ nodeΣT (σ ), form a partition of [D ]<ω . Take μ ∈ nodeΣT (σ ) and δ ∈ π (μ). Since π (μ) ⊂ [μ]<ω , by (3.1), we have that
Q (δ) ⊂
−1
Φ −1 M (d) ⊂ Φ M (t ) = S (μ). t ∈μ
d∈δ
Thus, L (μ) =
δ∈π (μ) Q (δ)
⊂ S (μ), and the construction of the sieve over the nodeΣT (σ ) is completed. Proceeding by induction on the levels of the tree Σ T , there exists a clopen-valued point-finite sieve L : Σ T → 2 X on X such that L (μ) ⊂ S (μ), for every μ ∈ Σ T . By Proposition 2.1, ΩR : B (ΣT ) → (C (Y ), τ V ) is a continuous perfect map. Then, define a mapping ψ : X → 2C (Y ) by
ψ(x) = ΩR 0L (x) = ΩR (β): β ∈ 0L (x) ,
x ∈ X.
According to [11, Propositions 5.1 and 5.2], 0L is nonempty-compact-valued and l.s.c., while, by [12, Corollary 4.6], it has a closed graph. Hence, ψ is l.s.c., and, by [12, Proposition 2.3], it has also a closed graph because ΩR is perfect. Since R is a (λ)-sieve on (C (Y ), τ V ), by [11, Lemma 7.1], ψ is a multi-selection for st[Φ, C ]. The implication (b) ⇒ (c) follows by Proposition 3.1, while (c) ⇒ (d) follows by Proposition 3.2. We complete the proof by showing that (d) ⇒ (a). Assume that X is as in (d), and U is a point-finite open cover of X . Endow U with the discrete topology, and define an l.s.c. mapping Φ : X → C (U ) by letting Φ(x) = {U ∈ U : x ∈ U }, x ∈ X . Since the Vietoris topology on C (U ) is discrete, (C (U ), τ V ) is actually the set [U ]<ω endowed with the discrete topology. Then, by (d), there exists a τ F -continuous mapping ψ : X → C ([U ]<ω ) such that σ ∩ Φ(x) = ∅, for every σ ∈ ψ(x) and x ∈ X . Since ψ is both l.s.c. and τ F -u.s.c., the family V σ = ψ −1 (σ ), σ ∈ [U ]<ω , is a point-finite clopen cover of X . Let
U σ = Φ −1 [σ ] =
Φ −1 (U ): U ∈ σ = σ ∈ U F,
σ ∈ [U ]<ω .
If x ∈ V σ , then σ ∈ ψ(x) and, therefore, σ ∩ Φ(x) = ∅. Hence, there is U ∈ σ , with U ∈ Φ(x), and we have that x ∈ Φ −1 (U ) ⊂ U σ . That is, V σ ⊂ U σ and, by Proposition 3.4, X is F -ultranormal. 2 By Theorem 3.3, we get the following consequence which furnishes the proof of Corollary 1.5. Corollary 3.6. For a space X , the following are equivalent: (a) X is F -ultrametacompact. (b) If Y is completely metrizable and Φ : X → F (Y ) is l.s.c., then the mapping st[Φ, C ] has an l.s.c. closed-graph multi-selection ψ : X → C (C (Y ), τ V ). (c) If Y is completely metrizable, then every l.s.c. mapping Φ : X → F (Y ) has an l.s.c. closed-graph multi-section ψ : X → C (C (Y ), τ V ). (d) If Y is completely metrizable, then every l.s.c. mapping Φ : X → F (Y ) has a τ F -continuous multi-section ψ : X → C (C ( Z ), τ V ). Proof. If X is a metacompact space, Y is a completely metrizable space and Φ : X → F (Y ) is l.s.c., by a result of Choban [6, Theorem 6.1], Φ has an l.s.c. multi-selection Ψ : X → C (Y ). Since st[Ψ, C ] is a multi-selection for st[Φ, C ], the implication (a) ⇒ (b) follows by Theorem 3.3. The implications (b) ⇒ (c) ⇒ (d) follow just like before by Propositions 3.1 and 3.2. As for (d) ⇒ (a), by Theorem 3.3, X is F -ultranormal. To show that X is metacompact, take an l.s.c. mapping Φ : X → F (Y ), where Y iscompletely metrizable. By (d), Φ has a τ F -continuous multi-section ψ : X → C (C ( Z ), τ V ). Define ϕ : X → C (Y ) by ϕ (x) = ψ(x), x ∈ X , see [17, Theorems 2.5]. Then, ϕ remains l.s.c., see [17, Theorems 5.7]. Thus, ϕ is an l.s.c. compactvalued section for Φ , and, by [12, Theorem 6.2], X is metacompact. 2 We conclude this section with an example showing that there are supermetacompact spaces which are not F ultrametacompact. Example 3.7. There exists a supermetacompact space X which is not F -ultrametacompact. Proof. We use virtually the same space X as given in [4, Example 4.2]. Namely, let X = [0, 1]2 \ {(0, 0)}, and for each x = 0, let H x = [0, 1] × {x} be the horizontal segment through the point (0, x), and V x = {x} × [0, 1] — the corresponding vertical one through the point (x, 0). Endow X with the topology in which all points (x, y ), x = 0 = y, are isolated, while the neighbourhoods of (0, x) and (x, 0), x = 0, consist of all but finitely many points of H x and, respectively, V x . Thus, if U ⊂ X
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is open and (0, x) ∈ U for some x = 0, then H x \ U is finite; similarly, V x \ U is finite if (x, 0) ∈ U . This implies that each open cover of X has a point-finite clopen refinement and, in particular, X is supermetacompact. However, X is not even F -ultranormal. Indeed, consider the point-finite open cover U of X , where
U = [0, 1] × (2−(n+1) , 2−n ], (2−(n+1) , 2−n ] × [0, 1]: n < ω .
Since U is countable, so is U F . We are only left to show that one cannot find a countable clopen refinement of U F . This follows from the observation that for any element U ∈ U F , any clopen subset W ⊂ U may contain only finitely many points of the type (0, x) or (x, 0) for x = 0. Consequently, one needs uncountably many such clopen subsets to cover the entire space X . 2 4. Sections, multi-sections and covering properties In this section, we discuss the relationship between sections and multi-sections of set-valued mappings, and their role in characterising covering properties. Our first result shows that, in the presence of τ V -continuity, sections and multi-sections are actually equivalent. Proposition 4.1. A mapping Φ : X → F (Y ) has a τ V -continuous multi-section ψ : X → C (C (Y ), τ V ) if and only if it has a τ V continuous section ϕ : X → C (Y ). In particular, a space X is superparacompact if and only if for every completely metrizable Y , every l.s.c. mapping Φ : X → F (Y ) has a τ V -continuous multi-section ψ : X → C (C (Y ), τ V ). Proof. If ϕ : X → C (Y ) is a τ V -continuous section for Φ , then ϕ : X → (C (Y ), τ V ) is continuous. In this case, the singletonvalued mapping ψ(x) = {ϕ (x)}, x ∈ X , is also τ V -continuous as a mapping from X to the compact subsets of (C (Y ), τ V ), and is a multi-section for Φ . Suppose that ψ : X → C (C (Y ), τ V ) is a τ V -continuous multi-section for Φ , and define a section ϕ : X → 2Y for Φ by ϕ (x) = ψ(x), x ∈ X . According to [17, Theorems 2.5 and 5.7], ϕ : X → C (Y ) and is τ V -continuous, which completes the first part of the proof. The second one now follows by Theorem 1.3. 2 Turning to the role of multi-sections in Theorem 3.3 and Corollary 3.6, let us explicitly mention that it is not clear if the simplistic approach used in Proposition 4.1 can be extended to τ F -continuous mappings. In fact, we have two different topologies, namely the Vietoris topology on C (Y ) and the Fell one on C (C (Y ), τ V ). In this regard, we have the following question. Question 1. Under what conditions on a space X , for every completely metrizable space Y , every l.s.c. mapping Φ : X → F (Y ) has a τ F -continuous section ϕ : X → C (Y )? We now turn to another interpretation of multi-sections. Proposition 4.2. For a metrizable space Y , a mapping Φ : X → F (Y ) has a τ F -continuous multi-section ψ : X → C (C (Y ), τ V ) if and only if there exists a metrizable space Z , a τ F -continuous mapping ϕ : X → C ( Z ) and a continuous perfect θ : Z → (C (Y ), τ V ) such that θ( z) ∩ Φ(x) = ∅ for every z ∈ ϕ (x) and x ∈ X . Proof. Let ψ : X → C (C (Y ), τ V ) be a τ F -continuous multi-section for Φ , Z be the space (C (Y ), τ V ), and θ : Z → C (Y ) be the identity map. Then, ϕ = ψ : X → C ( Z ) is τ F -continuous, while θ : Z → (C (Y ), τ V ) is a perfect continuous map (being a homeomorphism). Also, we have θ( z) ∩ Φ(x) = ∅ for every z ∈ ϕ (x) and x ∈ X . Conversely, let Z be a metrizable space, ϕ : X → C ( Z ) be τ F -continuous, and θ : Z → (C (Y ), τ V ) be a continuous perfect map such that θ(z) ∩ Φ(x) = ∅ for every z ∈ ϕ (x) and x ∈ X . Define ψ : X → C (C (Y ), τ V ) by ψ(x) = θ(ϕ (x)) = {θ( z): z ∈ ϕ (x)}, x ∈ X . Then, ψ remains l.s.c. as a composition of an l.s.c. mapping with a continuous one. Since θ is perfect, ψ is also τ F -u.s.c. If K ∈ ψ(x), then K = θ( z) for some z ∈ Z , and we have K ∩ Φ(x) = θ( z) ∩ Φ(x) = ∅. The proof is completed. 2 According to Corollary 1.5, we have the following immediate consequence. Corollary 4.3. A space X is F -ultrametacompact if and only if for every completely metrizable space Y and l.s.c. mapping Φ : X → F (Y ), there exists a metrizable space Z , a perfect continuous map θ : Z → (C (Y ), τ V ) and a τ F -continuous mapping ϕ : X → C ( Z ) such that θ(z) ∩ Φ(x) = ∅ for every z ∈ ϕ (x) and x ∈ X . This approach to multi-sections allows to extend some of the results for F -ultrametacompact spaces to supermetacompact spaces. Theorem 4.4. Let X be a supermetacompact space, Y be a completely metrizable space, and Φ : X → F (Y ) be an l.s.c. mapping. Then, there is a metrizable space Z , a τ F -continuous mapping ϕ : X → C ( Z ) and a τ V -continuous one θ : Z → C (Y ) such that θ(z) ∩ Φ(x) = ∅ for every z ∈ ϕ (x) and x ∈ X .
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Proof. Let ρ be a complete metric on Y , and M : T → 2Y be a nonempty-open-valued locally-finite sieve on Y with diamρ (M (t )) < 2−n , for every t ∈ T (n) and n < ω . Define a sieve R : Σ T → 2C (Y ) on (C (Y ), τ V ) by R (σ ) = M (σ ), for every σ ∈ Σ T . Next, define S : Σ T → 2 X by S (σ ) = (st[Φ, C ])−1 [R (σ )], σ ∈ Σ T . Then, by Proposition 2.3, S is an openvalued finitely additive sieve on X . We are going to construct a pruned tree ( D , ), a monotone map h : D → Σ T , and a point-finite clopen-valued sieve L : D → 2 X such that L is a multi-selection for S ◦ h. Here, h is monotone if h(s) ≺ h(t ) for every s, t ∈ D, with s ≺ t. Since {S (σ ): σ ∈ Σ T (0)} is a finitely additive open cover of X and X is supermetacompact, there exists a point-finite clopen cover { L d : d ∈ D 0 } of X and a map h0 : D 0 → Σ T (0) with L d ⊂ S (h0 (d)) for all d ∈ D 0 . We can proceed by induction. Namely, whenever p ∈ D 0 , consider the open cover {S (σ ) ∩ L p : σ ∈ nodeΣT (h0 ( p ))} of L p , which is finitely additive. Since L p is supermetacompact (being a clopen subset of X ), there exists a clopen point-finite cover { L d : d ∈ D ( p ,1) } of L p and a map h( p ,1) : D ( p ,1) → nodeΣT (h0 ( p )) such that L d ⊂ S (h( p ,1) (d)) ∩ L p for every d ∈ D ( p ,1) . Let D 1 be the disjoint union of the sets D ( p ,1) , p ∈ D 0 , and let h1 : D 1 → Σ T and π0 : D 1 → D 0 be defined by h1 D ( p ,1) = h( p ,1) , p ∈ D 0 , and respectively π0 (d) = p whenever d ∈ D ( p ,1) . This defines a point-finite clopen cover { L d : d ∈ D 1 } of X such
that L d ⊂ S (h1 (d)), d ∈ D 1 , and L p = { L d : d ∈ π0−1 ( p )}, p ∈ D 0 . Thus, by induction, for every n < ω we can define a set D n , maps hn : D n→ Σ T (n) and πn : D n+1 → D n , and a clopen point-finite cover { L d : d ∈ D n } of X such that L p ⊂ −1 S (hn ( p )) and L p = { L d : d ∈ πn ( p )}, p ∈ D n . Now, assuming that the sets D n , n < ω , are pairwise disjoint, we may define D = { D n : n < ω}, and a partial order “” on D such that ( D , ) is a tree for which D (n) = D n for each n < ω , and node D (d) = πn−1 (d) for each d ∈ D (n). Finally, define h : D → Σ T by h D n = hn , n < ω , and L : D → 2 X by L (d) = L d for d ∈ D. This completes the construction. Having already constructed these D, h and L , we conclude the proof by taking Z = B ( D ) and ϕ = 0L : X → C ( Z ). Then, ϕ is an l.s.c. closed-graph mapping (hence, τ F -continuous by Proposition 3.2), see the proof of (a) ⇒ (b) in Theorem 3.3. According to the construction above, h(β) ∈ B (Σ T ) for each β ∈ B ( D ). Moreover, the map h : B ( D ) → B (Σ T ) is continuous, see [11] and [14, Proposition 4.1]. By Proposition 2.1, the polar mapping ΩR : B (Σ T ) → C (Y ) is τ V -continuous, hence so is the composite mapping θ = ΩR ◦ h : Z → C (Y ). Since each
d∈β
R (h(d)) = θ(β), β ∈ B ( D ) = Z , is
a singleton of C (Y ) and L (d) ⊂ S (h(d)) = (st[Φ, C ])−1 [R (h(d))], d ∈ D, by [11, Lemma 7.1], the composite mapping θ ◦ ϕ : X → C (C (Y ), τ V ) is a multi-selection for st[Φ, C ]. Here, (θ ◦ ϕ )(x) = θ(ϕ (x)) ∈ C (C (Y ), τ V ), x ∈ X . This actually completes the proof. Indeed, if x ∈ X , then θ(ϕ (x)) ⊂ st[Φ, C ](x) and, therefore, θ( z) ∩ Φ(x) = ∅ for every z ∈ ϕ (x). 2 As the reader may observe, the difference between Corollary 4.3 and Theorem 4.4 is that in Corollary 4.3 the mapping
θ : Z → (C (Y ), τ V ) is also perfect. However, the authors were unable to show if the converse of Theorem 4.4 holds. We have only the following result for the special case of a locally compact range. Theorem 4.5. A space X is supermetacompact if and only if for every l.s.c. mapping Φ : X → F (Y ), where Y is a locally compact paracompact space, there exists a discrete space Z , a τ F -continuous mapping ϕ : X → C ( Z ) and a mapping θ : Z → C (Y ) such that θ(z) ∩ Φ(x) = ∅ for every z ∈ ϕ (x) and x ∈ X . Proof. Suppose that X is supermetacompact, Y is a locally compact paracompact space, and Φ : X → F (Y ) is l.s.c. According to a result of Choban and Michael [7], there exists an ultraparacompact space T , a perfect continuous surjection g : T → Y and a subset S ⊂ T such that g ( S ) = Y and g S is open. This implies that T is also locally compact. Hence, it has a discrete partition K consisting of nonempty compact subsets. Let Z = K F ⊂ C ( T ) be equipped with the Vietoris topology. Since {γ : γ ∈ [K ]<ω } is an open partition of Z consisting of singletons, Z is discrete. Define θ : Z → C (Y ) by θ( z) = g ( z), z ∈ Z . Since Φ is l.s.c., U = {Φ −1 [ g ( K ∩ S )]: K ∈ K } is an open cover of X . Since X is supermetaU F has a point-finite clopen refinement V . For every V ∈ V , fix a nonempty finite subset γ ( V ) ⊂ K with compact, − V ⊂ {Φ 1 [ g ( K )]: K ∈ γ ( V )}, and let z V = γ ( V ). Thus, z V ∈ Z and V ⊂ Φ −1 [ g ( z V )] = Φ −1 [θ( z V )]. Finally, define a mapping ϕ : X → 2 Z by ϕ (x) = { z V : x ∈ V }, x ∈ X . Since V is open and point-finite, ϕ is l.s.c. and finite-valued (hence, compact-valued as well). Since V is also closed and Z is discrete, ϕ is τ F -u.s.c. These Z , ϕ and g are as required. Indeed, if x ∈ X and z V ∈ ϕ (x), then x ∈ V ⊂ Φ −1 [θ( z V )], and therefore θ( z V ) ∩ Φ(x) = ∅. Conversely, take an open cover U of X , and endow it with the discrete topology. Next, define an l.s.c. mapping Φ : X → F (U ) by Φ(x) = {U ∈ U : x ∈ U }, x ∈ X . Since U is locally compact, there exists a discrete space Z , a τ F continuous mapping ϕ : X → C ( Z ) and a mapping θ : Z → C (U ) such that θ( z) ∩ Φ(x) = ∅, for every z ∈ ϕ (x) and x ∈ X . Then, V = {ϕ −1 ( z): z ∈ Z } is a clopen point-finite refinement of U F . Indeed, if z ∈ Z and x ∈ ϕ −1 ( z), then z ∈ ϕ (x) and, therefore, x ∈ Φ −1 [θ( z)] because θ( z) ∩ Φ(x) = ∅. That is,
ϕ −1 (z) ⊂ Φ −1 θ(z) =
θ(z) ∈ U F .
2
According to [3, Proposition 3.2], a Tychonoff space X is supermetacompact if and only if for every compact B ⊂ β X \ X , ˇ there exists a point-finite clopen cover W of X such that B ∩ W β X = ∅ for every W ∈ W . Here, β X is the Cech–Stone β X is the closure of W in β X . Theorem 4.5 can be regarded as a natural extension of this compactification of X , while W characterisation. To this end, let F∅ (Y ) = F (Y ) ∪ {∅} and C∅ (Y ) = C (Y ) ∪ {∅}. The Fell topology was considered originally on F∅ (Y ), and it was proved in [10] that (F∅ (Y ), τ F ) is always compact regardless of what is Y , [10, Lemma 1]. It was
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further proved in [10, Theorem 1] that (F∅ (Y ), τ F ) is compact and Hausdorff provided that Y is locally compact. In this result, Y is not assumed even to be T 1 . Now, we have the following result. Corollary 4.6. A Tychonoff space X is supermetacompact if and only if for every compact set B ⊂ β X \ X there exists a discrete space Z and a Fell-continuous mapping ψ : β X → F∅ ( Z ) such that ψ( X ) ⊂ C ( Z ) and ψ( B ) ⊂ {∅}. Proof. Suppose that X is a supermetacompact Tychonoff space and B ⊂ β X \ X is compact. For every point x ∈ X take a neighbourhood U (x) with U (x)β X ∩ B = ∅. Set U = {U (x): x ∈ X }, and endow it with the discrete topology. Next, define an l.s.c. mapping Φ : X → F (U ) by Φ(x) = {U ∈ U : x ∈ U }, x ∈ X . Then, by Theorem 4.5, there exists a discrete space Z , a τ F -continuous mapping ϕ : X → C ( Z ) and a mapping θ : Z → C (U ) such that θ( z) ∩ Φ(x) = ∅ for every z ∈ ϕ (x). Next, define ϕ∅ : X ∪ B → C∅ ( Z ) by ϕ∅ X = ϕ and ϕ∅ (x) = ∅ for x ∈ B. The basic neighbourhoods of ∅ in C∅ ( Z ) are given by the collection { Z \ K } = { S ∈ C∅ ( Z ): S ∩ K = ∅}, where K runs over the compact subsets of Z . Take a compact (i.e., finite)
subset K ⊂ Z . Then, ϕ −1 [ K ] is a clopen subset of X , and we have that ϕ −1 [ K ]β X ∩ B = ∅ because ϕ −1 [ K ] is contained in finitely many elements of U , see the proof of Theorem 4.5. This implies that ϕ∅ is τ F -continuous. Since (F∅ ( Z ), τ F ) is a compact Hausdorff space (as mentioned above), ϕ∅ can be extended to a τ F -continuous mapping ψ : β X → F∅ ( Z ). We have ψ( B ) = ϕ∅ ( B ) ⊂ {∅} and ψ( X ) = ϕ∅ ( X ) ⊂ C ( Z ), so these Z and ψ are as required. To show the converse, let B ⊂ β X \ X be compact and let ψ : β X → F∅ ( Z ) be τ F -continuous, where Z is a discrete space, ψ( X ) ⊂ C ( Z ) and ψ( B ) ⊂ {∅}. Then, each set ψ −1 ( z), z ∈ Z , is clopen in β X because { y ∈ β X: z ∈ / ψ( y )} is a neighbourhood of ∅ in (F∅ ( Z ), τ F ). Thus, {ψ −1 (z) ∩ X: z ∈ Z } is a point-finite, clopen cover of X because ψ( X ) ⊂ C ( Z ). Finally, since ψ( B ) ⊂ {∅} ⊂ F∅ ( Z ) \ C ( Z ), for every z ∈ Z we have that
ψ −1 ( z ) ∩ X β X ⊂ ψ −1 ( z ) ⊂ β X \ B . According to [3, Proposition 3.2], X is supermetacompact.
2
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