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Here is my approach to solve it but I get stuck at some point.

1) X is compact => f(X)=Y is compact (take an open cover of Y, its preimage covers X, but X is compact => exists a finite subcover and its image covers Y, thus Y is compact);

2) let C - a closed subset of X => C is compact (take an open cover of C, then it covers X in the union with the complement of C, which is open. X - compact, exists a finite subcover => there is a finite subcover covering C);

3) the same argument works for Y: all its closed subsets are compact.

And then, I get lost...There is one similar result, where Y is Hausdorff, but here we don't have this condition and know only that Y is compact. Is there something missing in the conditions? How should I proceed?

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This isn't true without the hypothesis that $Y$ is Hausdorff. For a very simple counterexample, let $X$ be any compact space whose topology is not indiscrete, let $Y$ have the same underlying set as $X$ but the indiscrete topology, and let $f:X\to Y$ be the identity map.

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