Let $(X,\tau_X)$ and $(Y,\tau_Y)$ be topological spaces, and let $f:X\to Y$ be continuous. If $\mathcal U \subset \tau_Y$ is an [open cover](/page/Open%20Cover) of $Y$, then the family
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\begin{align*}
f^{-1}\mathcal U := \{f^{-1}(U): U \in \mathcal U\}
\end{align*}