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 d_1 > 0, there is d_2 > 0 such that d(x, A) > d_1 implies
d(x) > d_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 ? > 0
the set {x ? E / d(x) > ?} is finite.
Keywords:Pseudo-metric space, Uniform space, Uniformly continuous function