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Čech complete nearness spaces

H. L. Bentley, Worthen N. Hunsaker (1992)

Commentationes Mathematicae Universitatis Carolinae

We study Čech complete and strongly Čech complete topological spaces, as well as extensions of topological spaces having these properties. Since these two types of completeness are defined by means of covering properties, it is quite natural that they should have a convenient formulation in the setting of nearness spaces and that in that setting these formulations should lead to new insights and results. Our objective here is to give an internal characterization of (and to study) those nearness...

Čech-completeness and ultracompleteness in “nice spaces”

Miguel López de Luna, Vladimir Vladimirovich Tkachuk (2002)

Commentationes Mathematicae Universitatis Carolinae

We prove that if X n is a union of n subspaces of pointwise countable type then the space X is of pointwise countable type. If X ω is a countable union of ultracomplete spaces, the space X ω is ultracomplete. We give, under CH, an example of a Čech-complete, countably compact and non-ultracomplete space, giving thus a partial answer to a question asked in [BY2].

Čech-Stone-like compactifications for general topological spaces

Miroslav Hušek (1992)

Commentationes Mathematicae Universitatis Carolinae

The problem whether every topological space X has a compactification Y such that every continuous mapping f from X into a compact space Z has a continuous extension from Y into Z is answered in the negative. For some spaces X such compactifications exist.

Centered-Lindelöfness versus star-Lindelöfness

Maddalena Bonanzinga, Mikhail Valerʹevich Matveev (2000)

Commentationes Mathematicae Universitatis Carolinae

We discuss various generalizations of the class of Lindelöf spaces and study the difference between two of these generalizations, the classes of star-Lindelöf and centered-Lindelöf spaces.

Central subsets of Urysohn universal spaces

Piotr Niemiec (2009)

Commentationes Mathematicae Universitatis Carolinae

A subset A of a metric space ( X , d ) is central iff for every Katětov map f : X → ℝ upper bounded by the diameter of X and any finite subset B of X there is x ∈ X such that f ( a ) = d ( x , a ) for each a ∈ A ∪ B . Central subsets of the Urysohn universal space 𝕌 (see introduction) are studied. It is proved that a metric space X is isometrically embeddable into 𝕌 as a central set iff X has the collinearity property. The Katětov maps of the real line are characterized.

Characterizations of Some Classes of Perfect Spaces in Terms of Continuous Selections Avoiding Supporting Sets

Takamitsu Yamauchi (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

Some kinds of perfect spaces, including paracompact perfectly normal spaces and collectionwise normal perfect spaces, are characterized in terms of continuous selections avoiding supporting sets. A necessary and sufficient condition on a domain space for a selection theorem of E. Michael [Fund. Math. 47 (1959), 173-178] to hold is also obtained.

Characterizing chainable, tree-like, and circle-like continua

Taras Banakh, Zdzisław Kosztołowicz, Sławomir Turek (2011)

Colloquium Mathematicae

We prove that a continuum X is tree-like (resp. circle-like, chainable) if and only if for each open cover 𝓤₄ = {U₁,U₂,U₃,U₄} of X there is a 𝓤₄-map f: X → Y onto a tree (resp. onto the circle, onto the interval). A continuum X is an acyclic curve if and only if for each open cover 𝓤₃ = {U₁,U₂,U₃} of X there is a 𝓤₃-map f: X → Y onto a tree (or the interval [0,1]).

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