Compactifications of Hausdorff spaces

Abstract

1. Introduction. In this paper, we describe methods of imbedding a Hausdorff space X in a compact space X so that each function in a given family of continuous functions on X has a continuous extension to X and the family of extensions separates the points of X -X. In particular, if X is completely regular but not locally compact, then we shall exhibit a non-Hausdorff compactification which contains X as an open subset and is bigger than the Stone-tech compactification of X. (Of course, every compactification of X is nonHausdorff if X is not completely regular.) We shall also show that the completion of a metric space M may be obtained as a subset of a compactification of M by a rather simple construction. By a compactification of a Hausdorff space X, we mean a compact space X which contains, as a dense subset, the image of X under a fixed homeomorphism f. We usually do not distinguish between X and f(X), and we say that X contains X as a dense subset. In what follows, X is always a noncompact Hausdorff space, AX denotes the closure of X-X in X, and a mapping is always a continuous function. If X is Hausdorff ,we say that X is a Hausdorff compactification of X. If X is not Hausdorff, however, we still assume that it satisfies