Free nuclear groups
Let \(\mathcal{C}\) be a non-empty class of abelian topological groups which is closed under forming arbitrary products and subgroups, and which contains all groups topologically isomorphic to a member of \(\mathcal{C}\). Given a topological space \(X\) and a free abelian group \(A(X)\) (with basis \( (\eta (x):x\in X)\), where \(\eta :X\rightarrow \eta (X)\) is a bijection), there exists a coarsest topology \(O_{A(X),\mathcal{C}}\) on \(A(X)\) such that for every \(G\in \mathcal{C}\) and every continuous mapping \(f:X\rightarrow G\), the unique homomorphism \(f':A(X)\rightarrow G\) which satisfies \(f'\circ \eta =f\) is continuous with respect to \(O_{A(X), \mathcal{C}}\). The free abelian group \(A(X)\) endowed with the topology \( O_{A(X),\mathcal{C}}\) will be denoted by \(A_{\mathcal{C}}(X)\) and, in case that \(A_{\mathcal{C}}(X)\in \mathcal{C}\), this group will be called free abelian group in \(\mathcal{C}\). One of the main results of this paper states that for a completely regular Hausdorff space \(X\), the free nuclear group \( A_{\mathcal{N}}(X)\) (i.e. the free abelian group in the class of all nuclear groups) exists. Two different descriptions of the topology on \(A_{\mathcal{N} }(X)\) are presented. The author also proves that if \(X\) is a product of metrizable spaces, then \(A(X)\) is nuclear if and only if \(X\) is discrete.
scientific article; zbMATH DE number 5080156