A compact subset $Y$ of a topological space $X$ is not necessarily closed.

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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?

Do you know any non-Hausdorff spaces? – Zev Chonoles Jan 05 '13 at 03:30 — Zev Chonoles, Jan 05 '13 at 03:30

score 8 · Accepted Answer · answered Jan 05 '13 at 03:37

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

answered Jan 05 '13 at 03:37

A compact subset $Y$ of a topological space $X$ is not necessarily closed.

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