Sequential definitions of compactness

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) is an element of 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. (C) 2007 Elsevier Ltd. All rights reserved.