Categorically proper homomorphisms of topological groups
\textit{D. N. Dikranjan} and \textit{V. V. Uspenskij} [J. Pure Appl. Algebra 126, No. 1--3, 149--168 (1998; Zbl 0887.22001)] call a (Hausdorff) topological group \(G\) \(c\)-compact if the projection \(G\times K\rightarrow K\) maps closed subgroups onto closed subgroups for every Hausdorff topological group \(K\). Surely compact groups are c-compact. Dikranjan and Uspenskij [loc. cit.] show that for soluble groups the converse is true. In general the converse is not true. This was proved by \textit{A. A. Klyachko} et al. [Topology Appl. 160, No. 16, 2104--2120 (2013; Zbl 1285.22003)]. In this paper, the authors study the related notion of \(c\)-properness in the category of (non-necessarily Haudorff) topological groups. In Section 2 categorically closed, proper and perfect maps are considered. In particular it is shown:\newline For a topological group \(G\) the following assertions are equivalent: (i) \(G\) is \(c\)-compact; (ii) every \(f:G\rightarrow H\) with \(H\) Hausdorff is \(c\)-proper; (iii) there is a \(c\)-proper \(f:G\rightarrow H\) with \(H\) \(c\)-compact. In Section 3 categorically and homomorphically complete objects and morphisms are studied. In particular it is shown:\newline For a topological group \(G\) the following assertions are equivalent: (i) \(G\) is \(c\)-complete; (ii) every \(f:G\rightarrow H\) with \(H\) Hausdorff is \(c\)-complete; (iii) there is a \(c\)-complete \(f:G\rightarrow H\) with \(H\) \(c\)-compact. These equivalences remain valid if ``\(c\)-complete'' is traded for ``\(h\)-complete'' everywhere. In Section 4 \(c\)-proper maps resp. \(h\)-complete maps are characterized by special filters. By using results from this section the following theorem is proved in Section 5:\newline Let \((f_i:G_i\rightarrow H_i)_{i\in I}\) be a family of \(c\)-proper (\(c\)-perfect; \(h\)-complete) maps with every \(H_i\) Hausdorff. Then their product \(\prod_{i\in I}f_i:\prod G_i\rightarrow \prod H_i\) is also \(c\)-proper (\(c\)-perfect; \(h\)-complete).\newline This theorem implies the following known result: The direct product of any family of \(c\)-compact groups is \(c\)-compact. Likewise, \(h\)-completeness is preserved under taking direct products of topological groups.
scientific article; zbMATH DE number 6641513
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