Timeline for Holes of a compact set in $\mathbb{R}^n$ that do not contain holes of a larger open set

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Feb 11, 2018 at 23:28	comment	added	erz		Just a slight distillation: let $X$ be a locally connected topological space, let $U\subset X$ be open, let $K\subset U$ be closed, let $W$ be a component of $U-K$ such that $\overline{W}\subset U$. Then $W$ is a component of $X-K$. Indeed, since $X$ is locally connected the components of open sets are open, and so every component of $U-K$ is open. Then the union $V$ of all the components other than $W$ is also open, and so $\overline{W}\cap V=\varnothing$. Hence, $\partial W\subset U-(W\cup V)=K$. Now $X-K\subset W\cup (X-\overline{W})$, and so $W$ is a component of $X-K$.
Feb 11, 2018 at 18:44	history	edited	Nate Eldredge	CC BY-SA 3.0	added 7 characters in body
Feb 11, 2018 at 18:39	history	answered	Nate Eldredge	CC BY-SA 3.0