# How to prove that if $f$ is continuous between topological spaces in $x \in X$ then it also continuous between metric spaces in $x \in X$?

A mapping $f$ between topological spaces $(X, \mathcal I_X)$ and $(Y, \mathcal I_Y)$ is continuous in $x \in X$ if $f^{-1}(V)$ is a neighborhood of $x$ for every neighborhood of $f(x)$. I want to show that this agree with the usual definition of continuity if $X$ and $Y$ are metric spaces and $\mathcal I_X$ and $\mathcal I_Y$ are the induced topologies of open sets.

I've shown that if $f$ is continuous between metric spaces in $x \in X$ then it is also continuous between topological spaces in $x \in X$.

However, how can I prove that if $f$ is continuous between topological spaces in $x \in X$ then it also continuous between metric spaces in $x \in X$ ?

I must find a $\delta > 0$ given $\epsilon > 0$, but the only thing I know is that $x \in U \subset \mathcal I_X$ and $f(x) \in U' \subset \mathcal I_Y$ both open ?

Let $B_\epsilon (f(x))$ be the open ball in $Y$ centered at $f(x)$ with radius $\epsilon$. By the topological definition of continuity, $f^{-1} (B_\epsilon (f(x))$ is open. Then remember how open set in defined in a metric space.