# Proving standard continuity using an alternative definition of homeomorphism

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.

The argument is quite short: Fix some $$x \in X$$ and let $$y=h(x)$$. Consider the open ball $$B_\epsilon(y)$$ around $$y \in Y$$ for some $$\epsilon>0$$. Since $$h_*$$ is a bijection, $$h_*^{-1}$$ is well-defined from $$\tau_2$$ to $$\tau_1$$ and so there exists an open $$U \in \tau_1$$ with $$x\in U$$ such that $$h_*^{-1}(B_{\epsilon}(y))=U$$. Since $$U$$ is open, it contains an open $$B_{\delta}(x)\subset U$$ for some $$\delta>0$$. Clearly $$h_*(B_{\delta}(x))\subset B_\epsilon(y)$$, which is essentially the definition of continuity you stated. The proof of continuity of the inverse is pretty much identical.