Products of infinite-dimensional spaces

Rohm, Dale M.

doi:10.1090/s0002-9939-1990-0946625-x

Cited by 14 publications

(5 citation statements)

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“…In his paper [13] Rohm asked when the product of two spaces with property S c (O, O) again has this property. In [5] and [13] the authors prove the following: Theorem 2 (Hattori, Yamada and Rohm).…”

Section: Summary Of Resultsmentioning

confidence: 99%

When does the Haver property imply selective screenability?

Babinkostova

2007

Topology and its Applications

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We point out that in metric spaces Haver's property is not equivalent to the property introduced by Addis and Gresham. We prove that they are equal when the space has the Hurewicz property. We prove several results about the preservation of Haver's property in products. We show that if a separable metric space has the Haver property, and the nth power has the Hurewicz property, then the nth power has the Addis-Gresham property. R. Pol showed earlier that this is not the case when the Hurewicz property is replaced by the weaker Menger property. We introduce new classes of weakly infinite dimensional spaces.In [6] Haver introduced for metric space (X, d) the following property: There is for each sequence (ε n : n < ∞) of positive real numbers a corresponding sequence (V n : n < ∞) where each V n is a pairwise disjoint family of open sets, each of diameter less than ε n , such that n<∞ V n is a cover of X. When a metric space has this property we say it has the Haver property. We consider the Haver property's relation to selection principles.Let A and B be given families of collections of subsets of some set S. Then the following symbols and statements define selection principles for the pair A, B:and {T m : m < ∞} ∈ B. • S fin (A, B): For each sequence (O m : m < ∞) of elements of A there is a sequence (T m : m < ∞) with each T m a finite subset of O m , and {T m : m < ∞} ∈ B. • S c (A, B): For each sequence (O m : m < ∞) of elements of A there is a sequence (T m : m < ∞) with each T m a pairwise disjoint family refining O m , and {T m : m < ∞} ∈ B. It is clear that S 1 (A, B) implies each of S fin (A, B) and S c (A, B). Even for very standard examples of A and B no other implications hold. For example, let S be a topological space, and let O denote the collection of open covers * Tel.

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Section: Summary Of Resultsmentioning

confidence: 99%

When does the Haver property imply selective screenability?

Babinkostova

2007

Topology and its Applications

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“…Since K p r is a C-space (cf. [8]) and Q is not, not all fibers of f are zero-dimensional (in fact not all of them are C-spaces), cf. [3, 5.4].…”

Section: Proposition 2 Suppose That X Is a Strongly Infinite-dimensiomentioning

confidence: 99%

On the weak and pointwise topologies in function spaces

Krupski

2015

RACSAM

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For a compact space K we denote by C w (K ) (C p (K )) the space of continuous real-valued functions on K endowed with the weak (pointwise) topology. In this paper we address the following basic question which seems to be open: Suppose that K is an infinite (metrizable) compact space. Can C w (K ) and C p (K ) be homeomorphic? We show that the answer is "no", provided K is an infinite compact metrizable C-space. In particular our proof works for any infinite compact metrizable finite-dimensional space K .

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“…A space X is called a C-space (or has property C) if for every sequencê x,^2, ... of open covers of X there exists a sequence %x, f/2, ... of families of pairwise disjoint open subsets of X , the union of which covers X , such that each member of ^, is contained in a member of % (see, for example, [3, §8] or [18] for this notion). Every countable-dimensional space has property C, and every space with property C is weakly infinite-dimensional.…”

Section: Introductionmentioning

confidence: 99%

“…Note that the weakly infinite-dimensional compactum which is not countabledimensional, constructed by R. Pol in [13], satisfies conditions (i) and (ii) (see [18,§3,first Corollary] for the proof of property (i)).…”

Section: Introductionmentioning

confidence: 99%