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Let $X$ a topological space. I have to prove it: If every non-empty open set of $X$ is dense in $X$, then if $C$ is an arbitrary own closed of $X$, then interior part of $C$ is empty. I tried to prove it but I don’t arrive at corrected conclusion.

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    $\begingroup$ What if $C=X$ ? $\endgroup$ – TheSilverDoe Apr 3 at 18:53
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If interior of $C$ is non empty, then it's a non-empty open set, then it is dense, then closure of $C$ is $X$, then closure of $C$ is $X$, then $C = X$ (as closed subset is closure of itself).

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