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On the completeness of topological groups. (Russian) Zbl 0536.22003

The author defines a suitable separable group topology \(\rho\) on a free algebraic group F(X) generated by a completely regular space X. We get the topological group \(F_{\rho}(X)\) such that the induced topology on X coincides with the original topology on X and the set \(F_ n(X)\) of all words of the length less than n is closed in \(F_{\rho}(X)\) for any natural n. A subset Y of a topological group H is called fine in H when for any neighbourhood V of unit there exists a neighbourhood W of unit such that \(xWx^{-1}\subseteq V\) for any \(x\in Y.\)
We introduce only two main results.
Theorem 1: Let X be a topological space and \(\tau\) be a group topology on F(X) agreeing with the topology of X. If X is fine in \(F_{\tau}(X)\) then the topology \(\rho\) of \(F_{\rho}(X)\) is stronger than \(\tau\).
Theorem 2: A topological group \(F_{\rho}(X)\) is complete in the sense of Weyl if and only the space X is complete in the sense of J. Dieudonné.
Reviewer: B.Smarda

MSC:

22A10 Analysis on general topological groups
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