Vol. 61, No. 1, 1975

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Contractibility of topological spaces onto metric spaces

Harold W. Martin

Vol. 61 (1975), No. 1, 209–217
Abstract

A space X is contractible onto a metric space provided there exists a one-to-one and continuous map from X onto a metric space. A variety of diagonal conditions are used to establish necessary and sufficient conditions in order that a space be contractible onto a metric space, and some metrization theorems are also obtained, e.g.: (i) a separable space is contractible onto a metric space if and only if has a zero-set diagonal; (ii) a space X is metrizable if and only if X has a compatible semi-metric which is uniformly continuous with respect to some product uniformity; (iii) a space X is contractible onto a metric space if and only if X has a sequence G1,G2, of open coverings which satisfies the following two conditions: (a) if xy, then there exists n such that y St (x,Gn); (b) if A,B Gn+1 and A B, then there exists C Gn with A B C; (iv) if Y is a perfectly normal space which is the pseudo-compact image of a space which is contractible onto a separable metric space, then Y is contractible onto a separable metric space.

Mathematical Subject Classification 2000
Primary: 54E35
Milestones
Received: 12 March 1975
Revised: 11 August 1975
Published: 1 November 1975
Authors
Harold W. Martin