Embedding the free topological group $$F(X^n)$$ into F(X)

Abstract

AbstractIn 1976, Nickolas showed that for each natural n, the free topological group $$F(X^n)$$ F ( X n ) is topologically isomorphic to a subgroup of F(X) provided X is a compact space or, more generally, a $$k_{\omega }$$ k ω -space. We complement the Nickolas’ embedding theorem by showing that it remains true for every topological space X such that all finite powers of X are pseudocompact. For example, all pseudocompact k-spaces enjoy this property. Also, we extend the embedding theorem to the class of $$NC_\omega $$ N C ω -spaces that includes, in particular, the $$k_\omega $$ k ω -spaces and the well-ordered spaces of ordinals $$[0, \alpha )$$ [ 0 , α ) , for every ordinal $$\alpha $$ α . Our results are quite sharp because we present a first example of a Tychonoff space Z such that F(Z) does not contain an isomorphic copy of the group $$F(Z^2)$$ F ( Z 2 ) . In addition, our space Z is countably compact, separable, and its square $$Z^2$$ Z 2 is not pseudocompact.