Compactifications of metric spaces

If X is a discrete topological space, the points of its Stone-Cech 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 (X) over tilde of a metric space in terms of the concept of near ultrafilters. We describe the topological space (X) over tilde and we investigate conditions under which (S) over tilde 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 (S) over tilde coincides with S-LUC.