Abstract:We consider $M$-mappings which include continuous mappings of spaces onto topological groups and continuous mappings of topological groups elsewhere. It is proved that if a space $X$ is an image of a product of Lindel\"of $\Sigma $-spaces under an $M$-mapping then every regular uncountable cardinal is a weak precaliber for $X$, and hence $ X$ has the Souslin property. An image $X$ of a Lindel\"of space under an $M$-mapping satisfies $cel_{\omega }X\le 2^{\omega }$. Every $M$-mapping takes a $\Sigma (\aleph _0)$-space to an $\aleph _0$-cellular space. In each of these results, the cellularity of the domain of an $M$-mapping can be arbitrarily large.
Keywords: $M$-mapping, topological group, Maltsev space, $\aleph _0$-cellularity
AMS Subject Classification: 54A25