A quantitative generalization of Prodanov–Stoyanov Theorem on minimal Abelian topological groups

Abstract

A topological group X is defined to have compact exponent if for some number n∈N the set {xn:x∈X} has compact closure in X. Any such number n will be called a compact exponent of X. Our principal result states that a complete Abelian topological group X has compact exponent (equal to n∈N) if and only if for any injective continuous homomorphism f:X→Y to a topological group Y and every y∈f(X)‾ there exists a positive number k (equal to n) such that yk∈f(X). This result has many interesting implications: (1) each minimal Abelian topological group is precompact (this is the famous Prodanov–Stoyanov Theorem); (2) an Abelian topological group is compact if and only if it is complete in each weaker Hausdorff group topology (this resolves a problem of Protasov and Zelenyuk of 2001); (3) an Abelian topological group X is complete and has compact exponent if and only if it is closed in each Hausdorff paratopological group containing X as a topological subgroup (this confirms a conjecture of Banakh and Ravsky of 2001).