Suppose $X,Y$ are metric spaces, let $A \subset X$ be a bounded subset of $X$ and $f: A \to Y$ to be a continuous bijection. Prove or disprove that $f^{-1}$ is continuous.
Remark: If each closed subset of $A$ is compact then $f$ would map closed sets to closed sets, which would then imply the continuity of $f^{-1}$. Then, how do we prove/disprove that $f$ is a closed map?
Best Answer
Let $X = \mathbb{R}$ with discrete metric and let $Y = \mathbb{R}$ with the usual metric. All maps $X \to Y$ are continuous. No maps $Y \to X$ are continuous.