Sections of the projection onto the monoid of the path-conncted components Let $M$ be a topological monoid and $A:=\pi_0(M)$ be the monoid consisting of its path connected components. Assume that $A$ is countable. We have a monoid homomorphism $\pi\colon M\longrightarrow A$. I would like to check that there exists at least a section of $\pi$, that is a monoid homomorphism $\sigma\colon A\longrightarrow M$ such that $\pi\circ\sigma=id_A$. 
I guess that in order to proceed one has to assume that the Axiom of Choice holds true. Any help?

This is false.  For instance, let $M=\bigcup_{n\in\mathbb{Z}}\{n\}\times[|n|,\infty)$, which is a topological monoid under coordinatewise addition.  The monoid $A$ of path-components is just $\mathbb{Z}$, with $\pi:M\to A$ being the first projection.  But there does not exist any nontrivial monoid homomorphism $A\to M$, since $M$ has no invertible elements besides $(0,0)$.
