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I'm trying to understand the proof of Urysohn's lemma (just to get some pespective).

This article says that "A topological space $X$ is normal iff for each closed subset $E$ of $X$ and each open set $W$ containing $E$, there exists an open set $U$ containing $E$ such that $\overline{U}\subset E$."

Shouldn't it be $\overline{U}\subseteq E$? What if $E$ is a clopen set?

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The author of that PDF is sloppy: he uses both $\subset$ and $\subseteq$ indiscriminately to mean $\subseteq$. And you’re quite right: here it has to be $\subseteq$ to accommodate the case in which $E=W$ is clopen.

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