Both compact and sequentially compact sets in abelian topological group

Abstract

We show that every abelian topological group contains many interesting sets which are both compact and sequentially compact. Then we can deduce some useful facts, e.g., (1) if G is a Hausdorff abelian topological group and μ : 2 N → G is countably additive, then the range μ ( 2 N ) = { μ ( A ) : A ⊆ N } is compact metrizable; (2) if X is a Hausdorff locally convex space and { x j } ⊂ X , then F = { ∑ j ∈ Δ x j : Δ ⊂ N , Δ is finite } is relatively compact in ( X , weak ) if and only if F is relatively compact in X, and if and only if F is relatively compact in ( X , F ( M ) ) where F ( M ) is the Dierolf topology which is the strongest 〈 X , X ′ 〉 -polar topology having the same subseries convergent series as the weak topology.