On the existence of free topological groups
For a class U of topological groups (such as for instance the class consisting of the circle group), and for a space X which is homeomorphic to a subspace of a product of factors selected from U, the authors construct a topological group G containing a copy of X such that any continuous function \(f: X\to H\) for a U-group H (that is, a group isomorphic to a closed subgroup of a product of U-groups) has a unique extension to a continuous group homomorphism \(G_ U\to H\). If G is itself isomorphic to a closed subgroup of a product of U-groups, then they call G the free U-group \(G_ U\). (They offer a variation involving a second class V of topological groups for the picking of H. This gives the (U,V)-free groups.) The authors show that for the class U of pseudocompact groups, a free U- group exists if and only if X is empty. They then turn to the case \(U=\{{\mathbb{R}}/{\mathbb{Z}}\}\), that is, to the case of free compact abelian groups. In this case one obtains FX, the character group Hom(C(X,\({\mathbb{R}}/{\mathbb{Z}}),{\mathbb{R}}/{\mathbb{Z}})\). The authors call PX the subgroup of all characters with countable range and define the group \(\xi\) X as the subgroup generated in \(F\beta\) X by the set \(X\cup P\beta X\). This turns out to be a (PA,CA)-free group for the class of pseudocompact, respectively, compact abelian groups such that X is embedded as a closed subspace. The authors construct other groups \(\eta\) X for certain spaces with properties very similar to those of \(\xi\) X but which allows no nonconstant continuous homomorphism \(\xi\) \(X\to \eta X\). The paper concludes with a list of unsolved problems.
scientific article; zbMATH DE number 4059722
scientific article