ON UNIFORM CONTINUITY AND COMPACTNESS IN PSEUDO METRIC SPACES
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## ON UNIFORM CONTINUITY AND COMPACTNESS IN PSEUDO METRIC SPACES

 Dr. S.M.Padhye1 and Ku. S.B. Tadam2 Associate Professor, Head of Dept. of Mathematics, Shri R.L.T. College of Science, Akola, Maharashtra, India Assistant Professor, Department of Mathematics, Shri R.L.T. College of Science, Akola, Maharashtra, India Related article at Pubmed, Scholar Google

### 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 Ã°ÂÂÂ¿1 > 0, there is Ã°ÂÂÂ¿2 > 0 such that d(x, A) > Ã°ÂÂÂ¿1 implies d(x) > Ã°ÂÂÂ¿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 Ã°ÂÂÂ > 0 the set {x Ã°ÂÂÂ E / d(x) > Ã°ÂÂÂ} is finite

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