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On quasi-uniform space valued semi-continuous functions

Tomasz Kubiak, María Angeles de Prada Vicente (2009)

Commentationes Mathematicae Universitatis Carolinae

F. van Gool [Comment. Math. Univ. Carolin. 33 (1992), 505–523] has introduced the concept of lower semicontinuity for functions with values in a quasi-uniform space ( R , 𝒰 ) . This note provides a purely topological view at the basic ideas of van Gool. The lower semicontinuity of van Gool appears to be just the continuity with respect to the topology T ( 𝒰 ) generated by the quasi-uniformity 𝒰 , so that many of his preparatory results become consequences of standard topological facts. In particular, when the order...

On relatively contractive relations in pairs of generalized uniform spaces.

Víctor M. Onieva Aleixandre, Javier Ruiz Fernández de Pinedo (1982)

Revista Matemática Hispanoamericana

J. C. Mathews and D. W. Curtis, [4], have introduced some structures which generalize structures of uniform types to the product of two sets, and they obtain a generalized version of Banach's contraction mapping theorem. In this note we prove that these structures are obtained from the usual analogues by means of a particular bijection; hence we do not have a meaningful generalization. For example, this bijection provides, from a result by A. S. Davies, [1], an analogue of Banach's well-known contraction...

On simple recognizing of bounded sets

Jan Hejcman (1997)

Commentationes Mathematicae Universitatis Carolinae

We characterize those uniform spaces and commutative topological groups the bounded subsets of which can be recognized by using only one uniformly continuous pseudometric.

On some questions in quasi-uniform topological spaces.

Jesús Ferrer Llopis, Valentín Gregori Gregori, Carmen Alegre Gil (1992)

Stochastica

Partial solution is given here respect to one open problem posed by P. Fletcher and W. F. Lindgren in their monography Quasi-uniform spaces.

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.

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