Semiuniform Spaces and Topological Homeomorphism Groups

A well-known sufficient condition that a group of homeomorphisms, $H$, from a topological space $X$ onto itself be a topological group relative to the topology of pointwise convergence is that $X$ be uniformizable and $H$ be equicontinuous. In this paper we prove an analogous condition in which the space $X$ is assumed to be only regular instead of completely regular (uniformizable). This is accomplished by means of the concepts of semiuniformity and semiequicontinuity introduced here.