Suppose that $X$ and $Y$ are topological spaces and that $X\times Y$ is a topological manifold. It seems that we can't conclude that $X$ or $Y$ are manifolds themselves (this question).
EDIT :Are there situations where we can say that $Y$ is a manifold given some $X$. For example, maybe if $X = (0,1)$, $(0,1)^{d}$, or $S^{1}$?
If so, any idea how one might prove these.
Thank you.