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**Aryth** We have a definition for continuity in terms of open sets of topologies, but I had a question about it. Here's the definition:

Let $\displaystyle (X,\Omega)$ and $\displaystyle (Y,\Theta)$ be topological spaces. A function $\displaystyle f: X \to Y$ is continuous relative to $\displaystyle \Omega$ and $\displaystyle \Theta$ provided $\displaystyle f^{-1}(U) \in \Omega$ for every $\displaystyle U \in \Theta$.

My question is this: Is $\displaystyle f^{-1} : Y \to X$ continuous relative to $\displaystyle \Theta$ and $\displaystyle \Omega$ provided $\displaystyle U \in \Theta$ for every $\displaystyle f^{-1}(U) \in \Omega$? Or is there something else you have to do to that sentence to make it true?