The natural mappings \(i_{n}\) and \(k\)-subspaces of free topological groups on metrizable spaces (Q1763595)

Let \(X\) be a Tychonoff space and \(F(X)\) the free topological group over \(X\), in the sense of Markov. For any natural \(n\), let \(F_n(X)\) be the closed subspace of \(F(X)\) formed by all words in \(F(X)\) whose reduced length does not exceed \(n\); \(j_n:F_n(X)\rightarrow F(X)\) the canonical inclusion and \(i_n\) the canonical map from the \(n\)th power of the topological sum \(X\oplus X^{-1}\oplus\{e\}\) onto \(F_n(X)\) (where \(e\) denotes the empty word). In this paper the author deals with the following list of properties: (1) \(F(X)\) is a \(k\)--space; (2) \(F(X)\) has the inductive limit topology with respect to the subspaces \(F_n(X)\); (3) \(F_n(X)\) is a \(k\)--space for every \(n\); (4) \(i_n\) is a quotient map for every \(n\); (5) \(X\) is either locally compact separable or discrete; (6) the topology on \(F(X)\) is the final topology with respect to the maps \(j_n\circ i_n.\) The equivalences \((1)\Leftrightarrow (5)\), \((3)\Leftrightarrow (4)\) and \((1)\Leftrightarrow(2)\), in the metrizable case, had been proved respectively by \textit{A. V. Arkhangel'skij, O. G. Okunev} and \textit{V. G. Pestov} [Topology Appl. 33, No.1, 63-76 (1989; Zbl 0689.54009)], the author [Topology Appl. 49, No.1, 75-94 (1993; Zbl 0817.54020)] and \textit{V. G. Pestov} and the author [Topology Appl. 98, No.1-3, 291-301 (1999; Zbl 0973.54034)]. In the first part of the paper the author proves the implication \((3)\Rightarrow(5),\) also in the metrizable case, hence showing that the first five properties are actually equivalent in that setting. In the third part it is shown that for a general \(X\), the property (6) holds if and only if (2) and (4) hold simultaneously. The author analyzes some of the natural analogues of these results for free Abelian topological groups. Using other results from the first and second above referred papers, an example for which the Abelian counterpart of \((3)\Rightarrow(1)\) does not hold is provided. On the other hand, the above mentioned result in the third part remains true in the Abelian context. The second part is devoted to a closer look at the case \(n=3\). It is proved that for metrizable \(X\), the properties (3') \(F_3(X)\) is a \(k\)--space; (4') \(i_3\) is a quotient map; their Abelian analogues, and the fact that either \(X\) is locally compact or the set of its nonisolated points is compact, are all equivalent.