Sequential definitions of compactness

Abstract

A subset F of a topological space is sequentially compact if any sequence x = ( x n ) of points in F has a convergent subsequence whose limit is in F . We say that a subset F of a topological group X is G -sequentially compact if any sequence x = ( x n ) of points in F has a convergent subsequence y such that G ( y ) ∈ F where G is an additive function from a subgroup of the group of all sequences of points in X . We investigate the impact of changing the definition of convergence of sequences on the structure of sequentially compactness of sets in the sense of G -sequential compactness. Sequential compactness is a special case of this generalization when G = lim .