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Find a topological space X and a compact subset A in X such that closure of A is not compact.

I first concluded that we must have X to be a non compact and a non Hausdorff space so that closure of A is not compact. I have been trying to look for an appropriate A such that closure of A is X and is hence not compact.

Asaf Karagila
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user186996
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HINT: Finite sets are always compact. But they can be dense, if you don't require $T_1$.

Asaf Karagila
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