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On the metric reflection of a pseudometric space in ZF

Horst Herrlich, Kyriakos Keremedis (2015)

Commentationes Mathematicae Universitatis Carolinae

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 one of the...

On the quantification of uniform properties

Robert Lowen, Bart Windels (1997)

Commentationes Mathematicae Universitatis Carolinae

Approach spaces ([4], [5]) turned out to be a natural setting for the quantification of topological properties. Thus a measure of compactness for approach spaces generalizing the well-known Kuratowski measure of non-compactness for metric spaces was defined ([3]). This article shows that approach uniformities (introduced in [6]) have the same advantage with respect to uniform concepts: they allow a nice quantification of uniform properties, such as total boundedness and completeness.

On three equivalences concerning Ponomarev-systems

Ying Ge (2006)

Archivum Mathematicum

Let { 𝒫 n } be a sequence of covers of a space X such that { s t ( x , 𝒫 n ) } is a network at x in X for each x X . For each n , let 𝒫 n = { P β : β Λ n } and Λ n be endowed the discrete topology. Put M = { b = ( β n ) Π n Λ n : { P β n } forms a network at some point x b i n X } and f : M X by choosing f ( b ) = x b for each b M . In this paper, we prove that f is a sequentially-quotient (resp. sequence-covering, compact-covering) mapping if and only if each 𝒫 n is a c s * -cover (resp. f c s -cover, c f p -cover) of X . As a consequence of this result, we prove that f is a sequentially-quotient, s -mapping if and only if it is...

On trivially semi-metrizable and D-completely regular mappings

F. Cammaroto, G. Nordo, B. A. Pasynkov (2002)

Bollettino dell'Unione Matematica Italiana

Trivially symmetrizable, trivially semi-metrizable and trivially D-completely regular mappings are defined. They are characterized as mappings parallel to symmetrizable, semi-metrizable and D-completely regular spaces correspondently. One shows that trivially D-completely regular mappings, i.e. submappings of fibrewise products of developable mappings coincide (up to homeomorphisms) with submappings of fibrewise products of semi-metrizable mappings.

On Weakly Measurable Functions

Szymon Żeberski (2005)

Bulletin of the Polish Academy of Sciences. Mathematics

We show that if T is an uncountable Polish space, 𝓧 is a metrizable space and f:T→ 𝓧 is a weakly Baire measurable function, then we can find a meagre set M ⊆ T such that f[T∖M] is a separable space. We also give an example showing that "metrizable" cannot be replaced by "normal".

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