Let $X$ and $Y$ be topological spaces. let $f: X \to Y$ and $g: Y \to X$. Assume that both $f$ and $g$ are continuous bijections. Can we say that $X$ and $Y$ are homeomorphic? If not are there assumptions we can place on the spaces so we do know this is true? (Like let $X$ and $Y$ be Hausdorff for instance.) I understand it isn't immediately obvious this should be true, but can't think of a counterexample.
Basically I am wondering if there is a Cantor–Schroeder–Bernstein Theorem for homeomorphisms.
I have been talking to several peers and have a possible simplification. Let $X = Y$; let $T$ and $S$ be two toplogies on $X$ and let $f:(X,T)\to(X,S)$ be the identity map. Then we can conclude $S$ is contained in $T$, or $T$ is finer than $S$ if you prefer.
Thanks for your thoughts.