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- W2645961192 abstract "A topological group $X$ is defined to have $compact$ $exponent$ if for some number $ninmathbb N$ the set ${x^n:xin X}$ 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 $ninmathbb N$) if and only if for any injective continuous homomorphism $f:Xto Y$ to a topological group $Y$ and every $yin bar{f(X)}$ there exists a positive number $k$ (equal to $n$) such that $y^kin f(X)$. 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)." @default.
- W2645961192 created "2017-06-30" @default.
- W2645961192 creator A5088614199 @default.
- W2645961192 date "2017-06-16" @default.
- W2645961192 modified "2023-09-27" @default.
- W2645961192 title "A quantitative generalization of Prodanov-Stoyanov Theorem on minimal Abelian topological groups" @default.
- W2645961192 hasPublicationYear "2017" @default.
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