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I've been struggling with providing examples of the following:

1) A continuous function $f$ and a connected set $E$ such that $f^{-1}(E)$ is not connected

2) A continuous function $g$ and a compact set $K$ such that $f^{-1}(K)$ is not compact

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    It suffices in both cases to take discrete spaces, and in 1) you can take finite discrete spaces.2012-02-16

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