Continuity_Test_using_Basis

Let $\left({X_1, \tau_1}\right)$ and $\left({X_2, \tau_2}\right)$ be topological spaces. Let $f: X_1 \to X_2$ be a mapping. Let $\mathcal B$ be an analytic basis for $\tau_2$. Suppose that $\forall B \in \mathcal B: f^{-1} \left({B}\right) \in \tau_1$ where $f^{-1} \left({B}\right)$ denotes the preimage of $B$ under $f$. Then $f$ is continuous.