elementary set theory - Intersection of images is empty implies intersection of preimages is empty.

Let $f:(X,\tau_X) \rightarrow (Y,\tau_Y)$ be a continuous function between topological spaces. Can you show that

$$U,V\in \tau_Y, ~U\cap V = \emptyset \implies f^{-1}(U)\cap f^{-1}(V) = \emptyset.$$
It is stated as a fact in a proof that path-connectedness implies connectedness.