Local connectedness in developable spaces

Abstract

A space is spherically connected if and only if it has an admissible semi-metric d such that ^-spheres of radius less than one are connected. It is shown that a developable space is locally connected if and only if it is spherically connected. A semi-metric space is -semi-metrizable if and only if it admits a semi-metric d such that d(A, B) > 0 whenever A and B are disjoint compact sets. It is shown that in the class of locally connected rim compact spaces, the l-semimetrizable spaces are precisely the developable ^-spaces. An example is given of a locally connected, locally compact Ksemi-metrizable Moore space which is not metrizable. 1* Introduction* A topological space is said to be rim compact