ON UNIFORM CONTINUITY AND COMPACTNESS IN PSEUDO METRIC SPACES

Abstract

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 𝛿1 > 0, there is 𝛿2 > 0 such that d(x, A) > 𝛿1 implies d(x) > 𝛿2 . 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 𝜖 > 0 the set {x 𝜖 E / d(x) > 𝜖} is finite