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On Equivalent Conditions for Compactness in Metric Spaces

Abstract

In this paper the comparison of two types of conditions (A) and (B) on a topological space is made. It is proved that the condition (A) is equivalent to (B) and normality of the topological space. Furthermore pre compactness is a necessary condition for (A) on a uniform space. Condition (A):- For any two disjoint closed sets at least one is compact. Condition (B):- For any two normally separable closed sets at least one is compact. Further it is proved that in metric spaces, these conditions (A) and (B) are equivalent to compactness

S. M. Padhye, Priti. P. Umarkar

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