Normality in free topological groups
The results of the paper are related to the following open question: Is the free topological group \(F(X)\) normal if all finite powers of \(X\) are normal? The author proves the following theorem giving necessary conditions for the normality of \(F(X)\) for a countably compact space \(X\): If a space \(X\) is countably compact, then its free topological group is normal if and only if all finite powers of \(X\) are normal and countably compact. The main result is contained in the Corollary on page 276: Corollary. If an HFD exists (in particular, if CH holds), then there is a topological space \(X\) all whose finite powers are normal (moreover, the countable power \(X^\omega\) is strongly collectionwise normal and has countable tightness, and the space \(X\) itself is countably compact) such that the free topological group \(F(X)\) is not normal. The notion of an HFD-set can be found in [\textit{A. Hajnal} and \textit{I. Juhász}, Fundam. Math. 81, 147--158 (1974; Zbl 0274.54002)] and [\textit{J. Roitman}, in: Handbook of set-theoretic topology. Amsterdam: Elsevier/North Holland. 295--326 (1984; Zbl 0594.54001)].
scientific article; zbMATH DE number 6968694