Sequential definitions of compactness

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