ON UNIFORM CONTINUITY AND COMPACTNESS IN PSEUDO METRIC SPACES

摘要

The Pseudo-metric spaces which have the property that all continuous real valued functions are uniformly continuous have been studied. It is proved that the following three conditions on pseudo-metric space X are equivalent a] Every continuous real valued function on X is uniformly continuous. b] Every sequence {xn} in X with lim d(xn) = 0 has a convergent subsequence. c] Set A is compact and for every e???1 0, there is e???2 0 such that d(x, A) e???1 implies d(x) e???2 . Here A = set of all limit points of X and d(x) = d(x, X- {x}) Further it is proved that in a pseudo-metric space X, a subset E of X is compact if and only if every continuous function f:E → R is uniformly continuous and for every e??? 0 the set {x e??? E / d(x) e???} is finite.