Extensions of proximity functions

Abstract:

Let ${P^ \ast }(X)$ be the algebra of bounded, real-valued proximity functions on a proximity space $(X,\delta )$, where $X$ is a dense subspace of a topological space $T$. In this paper we obtain several conditions which are equivalent to the following property: every member of ${P^ \ast }(X)$ has a continuous extension to $T$. Examples concerning these results are included, one of which shows that this extension property is distinct from ${C^ \ast }$-embedding.