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On the set-theoretic strength of the n-compactness of generalized Cantor cubes

Paul Howard, Eleftherios Tachtsis (2016)

Fundamenta Mathematicae

We investigate, in set theory without the Axiom of Choice , the set-theoretic strength of the statement Q(n): For every infinite set X, the Tychonoff product 2 X , where 2 = 0,1 has the discrete topology, is n-compact, where n = 2,3,4,5 (definitions are given in Section 1). We establish the following results: (1) For n = 3,4,5, Q(n) is, in (Zermelo-Fraenkel set theory minus ), equivalent to the Boolean Prime Ideal Theorem , whereas (2) Q(2) is strictly weaker than in set theory (Zermelo-Fraenkel set...

On uniformly locally compact quasi-uniform hyperspaces

Hans-Peter A. Künzi, Salvador Romaguera, M. A. Sánchez-Granero (2004)

Czechoslovak Mathematical Journal

We characterize those Tychonoff quasi-uniform spaces ( X , 𝒰 ) for which the Hausdorff-Bourbaki quasi-uniformity is uniformly locally compact on the family 𝒦 0 ( X ) of nonempty compact subsets of X . We deduce, among other results, that the Hausdorff-Bourbaki quasi-uniformity of the locally finite quasi-uniformity of a Tychonoff space X is uniformly locally compact on 𝒦 0 ( X ) if and only if X is paracompact and locally compact. We also introduce the notion of a co-uniformly locally compact quasi-uniform space and show...

On universality of countable and weak products of sigma hereditarily disconnected spaces

Taras Banakh, Robert Cauty (2001)

Fundamenta Mathematicae

Suppose a metrizable separable space Y is sigma hereditarily disconnected, i.e., it is a countable union of hereditarily disconnected subspaces. We prove that the countable power X ω of any subspace X ⊂ Y is not universal for the class ₂ of absolute G δ σ -sets; moreover, if Y is an absolute F σ δ -set, then X ω contains no closed topological copy of the Nagata space = W(I,ℙ); if Y is an absolute G δ -set, then X ω contains no closed copy of the Smirnov space σ = W(I,0). On the other hand, the countable power X ω of...

On universality of finite powers of locally path-connected meager spaces

Taras Banakh, Robert Cauty (2005)

Colloquium Mathematicae

It is shown that for every integer n the (2n+1)th power of any locally path-connected metrizable space of the first Baire category is 𝓐₁[n]-universal, i.e., contains a closed topological copy of each at most n-dimensional metrizable σ-compact space. Also a one-dimensional σ-compact absolute retract X is found such that the power X^{n+1} is 𝓐₁[n]-universal for every n.

On weak κ -metric

Bandlow, Ingo (1984)

Proceedings of the 12th Winter School on Abstract Analysis

On β-favorability of the strong Choquet game

László Zsilinszky (2011)

Colloquium Mathematicae

In the main result, partially answering a question of Telgársky, the following is proven: if X is a first countable R₀-space, then player β (i.e. the EMPTY player) has a winning strategy in the strong Choquet game on X if and only if X contains a nonempty W δ -subspace which is of the first category in itself.

On π -metrizable spaces, their continuous images and products

Derrick Stover (2009)

Commentationes Mathematicae Universitatis Carolinae

A space X is said to be π -metrizable if it has a σ -discrete π -base. The behavior of π -metrizable spaces under certain types of mappings is studied. In particular we characterize strongly d -separable spaces as those which are the image of a π -metrizable space under a perfect mapping. Each Tychonoff space can be represented as the image of a π -metrizable space under an open continuous mapping. A question posed by Arhangel’skii regarding if a π -metrizable topological group must be metrizable receives...

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