COMPACTIFICATIONS OF METRIC SPACES

Abstract

If $X$ is a discrete topological space, the points of its Stone-Čech compactification $\beta X$ can be regarded as ultrafilters on $X$, and this fact is a useful tool in analysing the properties of $\beta X$. The purpose of this paper is to describe the compactification $\tilde{X}$ of a metric space in terms of the concept of near ultrafilters. We describe the topological space $\tilde{X}$ and we investigate conditions under which $\tilde{S}$ will be a semigroup compactification if $S$ is a semigroup which has a metric. These conditions will always hold if the topology of $S$ is defined by an invariant metric, and in this case our compactification $\tilde{S}$ coincides with $S^{LUC}$.