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Weakly complete free topological groups. (English) Zbl 0979.54038

Let \(\mathcal C\) denote in this review one of the possibilities “sequentially”, “\(\omega\)”, “\(k\)” or “\(b\)” in the expressions \(\mathcal C\)-closed and \(\mathcal C\)-complete, where \(A\) is \(\omega\)-closed (or \(k\)-closed, or \(b\)-closed) in \(B\) if \(A\supset \overline C\) for all countable \(C\subset A\) (or \(A\cap C\) is closed in \(C\) for every compact (or functionally closed. resp.) \(C\subset B\)). A topological group is said to be \(\mathcal C\)-complete if it is \(\mathcal C\)-closed in its two-sided completion. It is shown that \(k\)-completeness and \(b\)-completeness coincide. The main results assert that for any Tikhonov space \(X\), the free topological group \(F(X)\) is \(\mathcal C\)-complete iff the free Abelian topological group \(A(X)\) is \(\mathcal C\)-complete iff \(X\) is \(\mathcal C\)-closed in its Dieudonné completion. Several examples to various relations between the above completions are given. For sequential completeness one may add to the main result: iff \(X\) is sequentially closed in \(\beta X\) iff the free precompact Abelian topological group over \(X\) is sequentially complete. (Question: Is it possible to remove “Abelian” from the last condition?).

MSC:

54H11 Topological groups (topological aspects)
54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
54D25 “\(P\)-minimal” and “\(P\)-closed” spaces
54A35 Consistency and independence results in general topology
54B30 Categorical methods in general topology
54D30 Compactness
54H13 Topological fields, rings, etc. (topological aspects)
22A05 Structure of general topological groups
22B05 General properties and structure of LCA groups
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