I was trying to understand the proof of a theorem from Munkres's book. He wants to prove that a compact subset $Y$ of an Hausdorff space $X$ is closed. He simply proves that, for all $x\in X-Y$, there is a neighborhood which is disjoint from $Y$. Why is this sufficient to conclude that $X-Y$ is open?
If there is a neighborhood disjoint from $Y$ for any $x\in X-Y$, $Y$ is closed.
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general-topology
2 Answers
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Because you can then write $X-Y$ as a union of open sets: Choose for every $x \in X-Y$ an open $U_x$ containing $x$, which is disioint form $Y$. Then $X-Y = \bigcup_{x\in X-Y}{U_x}$ is a union of opens, hence open.
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You can also prove this by showing that the closure of Y in X is open.