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ON UNIFORM CONTINUITY AND COMPACTNESS IN PSEUDO METRIC SPACES

Dr. S.M.Padhye and Ku. S.B. Tadam

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 ð�?¿1 > 0, there is ð�?¿2 > 0 such that d(x, A) > ð�?¿1 implies d(x) > ð�?¿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 ð�?�? > 0 the set {x ð�?�? E / d(x) > ð�?�?} is finite