Normality and N-factorizable topological groups

Abstract

By a result of A.A. Markov, every Tychonoff space is embeddable as a closed subspace into a Hausdorff topological group, so there is a wealth of Hausdorff topological groups that are not normal spaces. We introduce two very wide classes of topological groups (that are not necessarily normal spaces) as follows. A Hausdorff topological group G is called $N$-factorizable (resp., $Pc$-factorizable) if for every continuous real-valued function f on G, there exists a continuous homomorphism $\pi :G\to H$ onto a normal (resp., paracompact) topological group H such that $f=h\circ \pi$, for some continuous real-valued function h on H. We study the classes of $N$-factorizable and $Pc$-factorizable topological groups which contain all normal and, respectively, paracompact topological groups, in addition to all $R$-factorizable and $M$-factorizable topological groups. We show that every topological group is a quotient of a $Pc$-factorizable group.

As it turns out, the $N$-factorizable groups form a proper subclass of Hausdorff topological groups, whereas the $Pc$-factorizable groups are a proper subclass of the $N$-factorizable groups. The latter two classes of groups are closed when taking perfect homomorphic images. However, similar to normal spaces, the two classes are not finitely productive, even if the factors are ω-narrow groups. Several open problems are formulated.