Suppose $X$ is a topological space such that if $A$ and $B$ are homeomorphic open subsets of $X^n$ and $X^m$ then $n=m$, what can be said about $X$?
From a point-set point of view there doesn't seem to be much that can be said about $X$, for example invariance of domain fails for $\Bbb Q$, the Cantor space and finite discrete spaces, so by knowing that it holds for $X$ we can't conclude much about separation properties or compactness of $X$, however all of the counterexamples I know are totally disconnected so there might be some hope as far as connectedness is concerned.
From an algebraic topology point of view invariance of domain might give some more information on, say, homology of $X$, is that the case? More generally what can we say about $X$ if invariance of domain holds for it?