Properties of one-point completions of a noncompact metrizable space

Melvin Henriksen; Ludvík Janoš; Grant R. Woods

Displaying similar documents to “Properties of one-point completions of a noncompact metrizable space”

A non-archimedean Dugundji extension theorem

Jerzy Kąkol, Albert Kubzdela, Wiesƚaw Śliwa (2013)

Czechoslovak Mathematical Journal

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We prove a non-archimedean Dugundji extension theorem for the spaces $C^{*} (X, 𝕂)$ of continuous bounded functions on an ultranormal space $X$ with values in a non-archimedean non-trivially valued complete field $𝕂$ . Assuming that $𝕂$ is discretely valued and $Y$ is a closed subspace of $X$ we show that there exists an isometric linear extender $T : C^{*} (Y, 𝕂) \to C^{*} (X, 𝕂)$ if $X$ is collectionwise normal or $Y$ is Lindelöf or $𝕂$ is separable. We provide also a self contained proof of the known fact that any metrizable compact subspace $Y$ ...

Characterizing metric spaces whose hyperspaces are homeomorphic to ℓ₂

T. Banakh, R. Voytsitskyy (2008)

Colloquium Mathematicae

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It is shown that the hyperspace $C l d_{H} (X)$ (resp. $B d d_{H} (X)$ ) of non-empty closed (resp. closed and bounded) subsets of a metric space (X,d) is homeomorphic to ℓ₂ if and only if the completion X̅ of X is connected and locally connected, X is topologically complete and nowhere locally compact, and each subset (resp. each bounded subset) of X is totally bounded.

Wijsman hyperspaces of non-separable metric spaces

Rodrigo Hernández-Gutiérrez, Paul J. Szeptycki (2015)

Fundamenta Mathematicae

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Given a metric space ⟨X,ρ⟩, consider its hyperspace of closed sets CL(X) with the Wijsman topology $τ_{W (ρ)}$ . It is known that $⟨ C L (X), τ_{W (ρ)} ⟩$ is metrizable if and only if X is separable, and it is an open question by Di Maio and Meccariello whether this is equivalent to $⟨ C L (X), τ_{W (ρ)} ⟩$ being normal. We prove that if the weight of X is a regular uncountable cardinal and X is locally separable, then $⟨ C L (X), τ_{W (ρ)} ⟩$ is not normal. We also solve some questions by Cao, Junnila and Moors regarding isolated points in Wijsman hyperspaces. ...

Local/global uniform approximation of real-valued continuous functions

Anthony W. Hager (2011)

Commentationes Mathematicae Universitatis Carolinae

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For a Tychonoff space $X$ , $C (X)$ is the lattice-ordered group ( $l$ -group) of real-valued continuous functions on $X$ , and $C^{*} (X)$ is the sub- $l$ -group of bounded functions. A property that $X$ might have is (AP) whenever $G$ is a divisible sub- $l$ -group of $C^{*} (X)$ , containing the constant function 1, and separating points from closed sets in $X$ , then any function in $C (X)$ can be approximated uniformly over $X$ by functions which are locally in $G$ . The vector lattice version of the Stone-Weierstrass Theorem is more-or-less...

On the metric reflection of a pseudometric space in ZF

Horst Herrlich, Kyriakos Keremedis (2015)

Commentationes Mathematicae Universitatis Carolinae

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We show: (i) The countable axiom of choice $𝐂𝐀𝐂$ is equivalent to each one of the statements: (a) a pseudometric space is sequentially compact iff its metric reflection is sequentially compact, (b) a pseudometric space is complete iff its metric reflection is complete. (ii) The countable multiple choice axiom $𝐂𝐌𝐂$ is equivalent to the statement: (a) a pseudometric space is Weierstrass-compact iff its metric reflection is Weierstrass-compact. (iii) The axiom of choice $𝐀𝐂$ is equivalent to each...

Some versions of second countability of metric spaces in ZF and their role to compactness

Kyriakos Keremedis (2018)

Commentationes Mathematicae Universitatis Carolinae

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In the realm of metric spaces we show in ZF that: (i) A metric space is compact if and only if it is countably compact and for every $ε > 0$ , every cover by open balls of radius $ε$ has a countable subcover. (ii) Every second countable metric space has a countable base consisting of open balls if and only if the axiom of countable choice restricted to subsets of $ℝ$ holds true. (iii) A countably compact metric space is separable if and only if it is second countable.