Consider the following definition for a homeomorphism $h$ between metric spaces $(X,d_1)$ and $(Y,d_2)$, with metric topologies $\tau_1$ and $\tau_2$ respectively:
- Definition: $h: X \rightarrow Y$ is a homeomorphism if $h$ is a bijection between $X$ and $Y$, and the induced map $h_*$ from $\tau_1$ to $\tau_2$ (which is just $h$ redefined to take open sets as input) is a bijection as well. This definition can be found, for example, in this post.
I would like to show that this definition implies "standard continuity" for $h$; that is, that given any $\epsilon > 0$, there exists a $\delta > 0$ such that $d_1(x_1,x_2) < \delta$ implies $d_2(h(x_1),h(x_2)) < \epsilon$ for any such elements $x_1,x_2 \in X$.
Ideally, I would like the argument to be as direct as possible, i.e. avoiding statements that prove $h$ is continuous according to an alternate equivalent definition, which then implies this.