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On the set function

Sergio Macías (2024)

Commentationes Mathematicae Universitatis Carolinae

Inspired by the work that Professor Janusz R. Prajs did on homogeneous metric continua in his paper (2010) and the version of his work for Hausdorff continua with the uniform property of Effros done by this author, we introduce a new set function, , and present properties of it.

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.

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