2.8 Practice Problems
Problem 2.8.1. Let \(f:X\rightarrow Y\) be a map of topological spaces. Prove that \(f\) is continuous if and only if \(f^{-1}(F)\) is closed in \(X\) for every closed \(F\subseteq Y\).
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Solution. The key is that preimages commute with complements: \[f^{-1}\left (Y\setminus F\right ) = X\setminus f^{-1}(F).\]
Suppose \(f\) is continuous and \(F\) is closed in \(Y\). Then \(Y\setminus F\) is open, so \(f^{-1}(Y\setminus F) = X\setminus f^{-1}(F)\) is open, and therefore \(f^{-1}(F)\) is closed.
Conversely, suppose preimages of closed sets are closed and let \(V\subseteq Y\) be open. Then \(Y\setminus V\) is closed, so \(X\setminus f^{-1}(V)\) is closed, and hence \(f^{-1}(V)\) is open. So \(f\) is continuous.
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