Free topological groups and inductive limits
Let \(X\) be a Tychonoff space, and let \(F(X)\) be the (Markov) free topological group over \(X\). \(F_ n(X)\) consists of all words in \(F(X)\) whose reduced length does not exceed \(n\). In this very interesting paper, the author proves among other things the following results: (1) If \(X^ n\) is pseudocompact for some \(n \in N^ +\), then \(\beta (F_ n(X))\) is homeomorphic to \(F_ n(\beta X)\), where \(\beta\) denotes the Čech- Stone compactification. The converse is also true. (2) Let \(X\) be a pseudocompact space, and let \(F(X)\) be an inductive limit of its subspaces \(F_ n(X)\), \(n \in N\). Then all finite powers of \(X\) are normal and countably compact. Note that the author showed the converse of this result [in Mat. Zametki 41, 400-405 (1987; Zbl 0622.22001)]. In section 4 an example is constructed showing that the inductive limit property of \(F(X)\) is destroyed by the free topological sum operation. Finally in section 5 three problems are stated, and a brief survey of results is given concerning these problems.
scientific article; zbMATH DE number 695880
scientific article