Let $X$ and $Y$ be two topological spaces. Let $f : X \to Y$ and $g: Y \to X$, such that
- $f$ and $g$ are surjective;
- $f$ and $g$ are continous.
Does this imply that $X$ and $Y$ are homeomorphic?
It seems similar to Bernstein's theorem in set theory, and many of the topological properties like compactness, connectedness, etc are getting preserved.
Any help would be appreciated.