I'm going through Bishop and Goldberg's "Tensor Analysis on Manifolds" right now and I'm stuck in Chapter 0. They give a proof of the statement "A compact subset of a Hausdorff space is closed" that I can't seem to wrap my head around. I'm reprinting the proof here:(adsbygoogle = window.adsbygoogle || []).push({});

"Suppose that [itex]A[/itex] is a compact subset in a Hausdorff space [itex]X[/itex] and [itex]A \neq A^-[/itex] (where [itex]A^-[/itex] denotes the closure of [itex]A[/itex]), so there is an [itex]x \in A^- - A[/itex]. For every [itex]a \in A[/itex] there are open sets [itex]G_a, G^x_a[/itex] such that [itex]G_a \cap G^x_a = \emptyset , a \in G_a[/itex], and [itex]x \in G^x_a[/itex], because [itex]X[/itex] is Hausdorff. Then [itex]\{G_a|a \in A\}[/itex] is an open covering of [itex]A[/itex], so there is a finite subcovering [itex]\{G_a|a \in J\}[/itex], where [itex]J[/itex] is a finite subset of [itex]A[/itex]."

I'm fine with everything up to this point, but the next sentence loses me:

"But then [itex]\bigcap_{a \in J} G^x_a[/itex] is a neighborhood of [itex]x[/itex] which does not meet [itex]\bigcup_{a \in J} G_a \supset A[/itex], so [itex]x[/itex] cannot be in [itex]A^-[/itex], a contradiction."

The authors have previously defined the closure of a set [itex]A[/itex] as the intersection of all closed sets containing [itex]A[/itex]. I get that [itex]\bigcap_{a \in J} G^x_a[/itex] is not a subset of [itex]A[/itex], but I don't understand why that implies that [itex]x \notin A^-[/itex].

[EDIT]: I recalled another section earlier in the book saying that [itex]x \in A^-[/itex] iff every neighborhood of [itex]x[/itex] intersects [itex]A[/itex], which makes the last sentence in question trivial.

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# Question about proof from Bishop & Goldberg

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