Let $X$ and $Y$ be compact metric spaces. Let $f:X \to Y$ be a Borel measurable map and suppose that $T:X \to X$ is a homeomorphism. Can one change the topology on $X$ such that
- $X$ is still a compact metrizable space, with the same Borel sets.
- The map $T$ remains continuous.
- The map $f:X \to Y$ is continuous.
Note: If $Y$ is not compact, we can find counterexamples by taking $f$ to be an unbounded function. The answer is YES if we just want complete metrizability of the new topology on $X$, instead of compactness.