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 $n\in\mathbb N$ the set $\{x^n:x\in X\}$ 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 $n\in\mathbb N$) if and only if for any injective continuous homomorphism $f:X\to Y$ to a topological group $Y$ and every $y\in \bar{f(X)}$ there exists a positive number $k$ (equal to $n$) such that $y^k\in f(X)$. This result has many interesting implications: (1) an Abelian topological group is compact if and only if it is complete in each weaker Hausdorff group topology; (2) each minimal Abelian topological group is precompact (this is the famous Prodanov-Stoyanov Theorem); (3) a 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 topoloical subgroup (this confirms an old conjecture of Banakh and Ravsky).