On Fan-Gottesman compactification and P. S. Alexsandrov's problems

Abstract

As a generalization of Freudenthal’s compactification of rim-compact Hausdorff spaces [S], Fan and Gottesman introduced the concept of normal base and have constructed a compactification for any regular space with a normal base [S]. The Fan-Gottesman compactification includes Wallman compactification of normal spaces and Aleksandrov one-point compactification of locally compact spaces as particular cases. The results in the present paper are mainly consequences of Fan-Gottesman Theorem [5, Theorem 11; other concepts employed are Aleksandrov compactification ciT of a completely regular space T [I], [7], W 11 a man H-closed extension w(X) of a Hausdorff space X, and the maximal binding families relative to all open sets in a space [5]. A new concept called pseudonormality, analogous to but weaker than normality, is introduced. A regular pseudonormal space is completely regular but the converse is untrue. There are pseudonormal spaces that are not normal, namely, the well-known Sorgenfrey plank, Tychonoff plank, and the Moore (Niemytzki) plane. As a real-valued continuous function f (bounded or unbounded) on a Hausdorff space X can be continuously extended over w(X), denoted by f, we will show that f has a unique extension [ on the collection X** of maximal binding families in X in some sense described below. A Hausdorff space X is weakly periphery-compact if and only if continuous functions j on w(X) separate points in w(X); and a regular space X is pseudonormal if and only if all Jon X* * separate points in X**. A Hausdorff space is completely regular if and only if for each pair (x, y), x E X, y E X**, and x # y, there is a continuous function f on X such that f(x) #J(y). A new construction for the Stone-Tech compactification j3T of a completely regular space T, based on the Wallman H-closed extension W(T), will be given and hence a new proof for the Stone-Tech extension theorem. The