Continuity of a Map Between Metric Spaces via Preimages of Open Sets

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces. Let $\mathcal{T}_{d_X}$ be the collection of all subsets of $X$ that are open in $(X,d_X)$ and let $\mathcal{T}_{d_Y}$ be the collection of all subsets of $Y$ open in $(Y,d_Y)$; both are topologies by Metric Open Sets Form a Topology. Let $f:X\to Y$.

Then $f$ is continuous on $X$ if and only if for every $V\in\mathcal{T}_{d_Y}$ the preimage

$$f^{-1}(V)=\{x\in X:\ f(x)\in V\}$$

belongs to $\mathcal{T}_{d_X}$.