A covering property for metric spaces

Haver, William E.

doi:10.1007/bfb0064016

Cited by 54 publications

(36 citation statements)

References 6 publications

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“…Recall that a space X is a C-space [1] if for any sequence {ω n : n ∈ N} of open covers of X there exists a sequence {γ n : n ∈ N} of open disjoint families in X such that each γ n refines ω n and {γ n : n ∈ N} covers X. The C-space property was introduced by Haver [15] for compact metric spaces, and Addis and Gresham [1] extended Haver's definition to more general spaces. All countable-dimensional metrizable spaces (spaces which are countable unions of finite-dimensional subsets), in particular all finite-dimensional ones, form a proper subclass of the class of C-spaces because there exists a metric C-compactum which is not countable-dimensional [27].…”

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confidence: 99%

On dimensionally restricted maps

Tuncali

Valov

2002

Fund. Math.

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Abstract. Let f : X → Y be a closed n-dimensional surjective map of metrizable spaces. It is shown that if Y is a C-space, then: (1) the set of all maps g :These are extensions of theorems by Pasynkov and Toruńczyk, respectively, obtained for finite-dimensional spaces. A generalization of a result due to Dranishnikov and Uspenskij about extensional dimension is also established. Introduction.All spaces are assumed to be completely regular and all maps continuous. This paper is concerned with the following two results. The first one was proved by Pasynkov [25] (see [24] for noncompact versions) and the second one by Toruńczyk [31]:

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confidence: 99%

On dimensionally restricted maps

Tuncali

Valov

2002

Fund. Math.

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“…[1]). More generally, one may only require that each 4>"(C) is a C-space (see [7]). This follows from the fact that Lemma 2 holds when A is a subspace of C with the weaker assumption that each An is a (not necessarily closed) C-space.…”

Section: Proof the Banach-steinhausmentioning

confidence: 99%

On Extending Mappings into Nonlocally Convex Linear Metric Spaces

Dobrowolski¹

1985

Proceedings of the American Mathematical Society

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Abstract.It is proved that the following spaces are absolute retracts: every F-space with a Schauder basis and certain function spaces along with their subgroups of integer-valued elements. It is also observed that for every o-compact convex set, the absolute extension property for compacta implies the AR-property.1. Introduction. The purpose of this paper is to provide new examples of infinite-dimensional ANRs. Detecting the ANR-property of convex subsets of nonlocally convex metric linear spaces and topological groups is of great importance. For example, the topological classification of these spaces, given recently in [4, 5 and 3], required the ANR-property. We prove that the following spaces are absolute retracts: (1) every complete metric linear space (= F-space) with a Schauder basis, (2) certain function spaces which include Lp (p 5= 0) and Orlicz spaces, and (3) additive subgroups consisting of all integer-valued functions in certain function spaces. Consequently, each of these spaces, when complete and separable, is homeomorphic to a Hubert space [4, 5]. The argument used in verifying the AR-property of the above examples is also employed to show that the AR and the AE(fé') (absolute extension property for compacta) properties coincide for a-compact convex sets. This enables us to find a dense convex topological copy of 2, the linear span of the Hilbert cube in the Hilbert space l2, in every separable infinite-dimensional complete convex set.Our approach is very elementary and mostly involves the natural equiconnected structures of convex sets and contractible groups. We also employ the admissibility

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“…The C-space property was originally defined by W. Haver [9] for compact metric spaces. Later on, Addis and Gresham [1] reformulated Haver's definition for arbitrary spaces: A space X has property C (or X is a C-space) if for every sequence {W n : n < ω} of open covers of X there exists a sequence {V n : n < ω} of pairwise disjoint open families in X such that each V n refines W n , n < ω, and {V n : n < ω} is a cover of X.…”

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Open maps having the Bula property

Gutev

Valov

2009

Fund. Math.

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Abstract. An open continuous map f from a space X onto a paracompact C-space Y admits two disjoint closed sets F0, F1 ⊂ X with f (F0) = Y = f (F1), provided all fibers of f are infinite and C * -embedded in X. Applications are given to the existence of "disjoint" usco multiselections of set-valued l.s.c. mappings defined on paracompact C-spaces, and to special type of factorizations of open continuous maps from metrizable spaces onto paracompact C-spaces. This settles several open questions.

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A covering property for metric spaces

Cited by 54 publications

References 6 publications

On dimensionally restricted maps

On dimensionally restricted maps

On Extending Mappings into Nonlocally Convex Linear Metric Spaces

Open maps having the Bula property

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