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Possible Duplicate:
Compact sets are closed?

We know that if $X$ is Hausdorff, then a compact subset $Y$ of $X$ must be closed. Without the assumption, this claim is not true. But can you come up with a counterexample?

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marked as duplicate by Austin Mohr, amWhy, Potato, Alexander Gruber, Asaf Karagila Jan 5 '13 at 9:48

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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Do you know any non-Hausdorff spaces? – Zev Chonoles Jan 5 '13 at 3:30
up vote 6 down vote accepted

Give any set with at least two elements the indiscrete topology. Then any nonempty proper subset is compact but not closed.

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