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A note on quasi-uniform continuity

Published online by Cambridge University Press:  17 April 2009

Panayotis Th. Lambrinos
Panayotis Th. Lambrinos
Affiliation:
Department of Mathematics, AristotleUniversity of Thessaloniki, Thessaloniki, Greece.
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Abstract

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It is shown that:

(i) a continuous function from a compact quasi-uniform space into a quasi-uniform space is not necessarily quasiuniformly continuous;

(ii) if the range of the function is a uniform space, the function will be necessarily quasi-uniformly continuous.

(This contradicts an example in the literature and generalizes a classical result.) Finally, a generalization of (ii) is given by means of a suitable boundedness notion.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 8 , Issue 3 , June 1973 , pp. 389 - 392
Copyright
Copyright © Australian Mathematical Society 1973

References

[1]Bourbaki, N., Éléments de mathématique, Livre III, Topologie générale, 2e édition, Chapitres I et II (Hermann, Paris, 1951).Google Scholar
[2]Fletcher, P., “On totally bounded quasi-uniform spaces”, Arch. Math. 21 (1970), 396–401.CrossRefGoogle Scholar
[3]Kelley, John L., General topology (Van Nostrand, Toronto, New York, London, 1955).Google Scholar
[4]Lambrinos, Panayotis, “A topological notion of boundedness”, submitted.Google Scholar
[5]Murdeshwar, M.G. and Naimpally, S.A., Quasi-uniform topological spaces (Noordhoff, Series A: Preprints of Research Papers, No. 4, Vol. 2, 1966).Google Scholar