ISSN ONLINE(2319-8753)PRINT(2347-6710)

All submissions of the EM system will be redirected to Online Manuscript Submission System. Authors are requested to submit articles directly to Online Manuscript Submission System of respective journal.

ON UNIFORM CONTINUITY AND COMPACTNESS IN PSEUDO METRIC SPACES

Visit for more related articles at International Journal of Innovative Research in Science, Engineering and Technology

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