Vol. 44, No. 1, 1973

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Proximity approach to semi-metric and developable spaces

Mani Gagrat and Somashekhar Amrith Naimpally

Vol. 44 (1973), No. 1, 93–105

In this paper we study semi-metric and developable spaces via generalized proximities and uniformities. We find sufficient conditions for a compatible semi-metric d on a space X to induce a Lodato proximity and also study the effects on a space X when d satisfies various weaker forms of continuity. We present two new characterizations of developable spaces, one of which reads: A T1-space is developable if and only if it has a compatible upper semi-continuous semimetric. We give improved versions of two known metrization theorems. Finally, we generalize the concepts: T1-map, uniform map, completely uniform map, pseudo-open map, etc., to apply to proximity spaces and improve some of the known results: for example, an open uniform image of a developable space is developable.

Received: 15 July 1971
Revised: 1 May 1972
Published: 1 January 1973
Mani Gagrat
Somashekhar Amrith Naimpally