Let $f:(X,\tau_1) \to(Y,\tau_2)$ be an injective and surjective continuous function. If $X$ is compact with respect to $\tau_1$ and $Y$ is Hausdorff with respect to $\tau_2$ then how can we show that $f$ is a Homeomorphism?
I know that every bijective bi-continuous mappings are homeomorphic. Here it is given that this mapping is bijective and continuous, how can I show that the inverse map is continuous?