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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.

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