నైరూప్య

On Equivalent Conditions for Compactness in Metric Spaces

S. M. Padhye, Priti. P. Umarkar

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