Continuous Mapping is Continuous on Induced Topological Spaces

Theorem
Let $$M_1 = \left\{{A_1, d_1}\right\}$$ and $$M_2 = \left\{{A_2, d_2}\right\}$$ be metric spaces.

Let $$\vartheta_{\left\{{A_1, d_1}\right\}}$$ and $$\vartheta_{\left\{{A_2, d_2}\right\}}$$ be the topologies induced by the metrics $$d_1$$ and $$d_2$$.

Let $$T_1 = \left\{{A_1, \vartheta_{\left\{{A_1, d_1}\right\}}}\right\}$$ and $$T_2 = \left\{{A_2, \vartheta_{\left\{{A_2, d_2}\right\}}}\right\}$$ be the resulting topological spaces.

Let $$f: A_1 \to A_2$$ be $\left({d_1, d_2}\right)$-continuous.

Then $$f$$ is also $\left({\vartheta_{\left\{{A_1, d_1}\right\}}, \vartheta_{\left\{{A_2, d_2}\right\}}}\right)$-continuous.

Proof
Follows directly from:
 * the open set definition of continuity on a metric space;
 * the definition of continuity on a topological space.