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In Basic Topology, Armstrong states, "a topological space is compact if every open cover of X has a finite subcover." But, I'm mildly confused how this doesn't translate to the empty set being a finite subcover itself.

First, let X be a compact space and let U be an open cover of X. Thus, there exists a finite subcover of X, say $\{x_1, ..., x_n\}$. But then, this itself is an open cover of X, since a set of open sets whose union is the space is an open cover. Thus, there exists some element $x_i$ s.t. $$\bigcup\{x_1, \ldots, \widehat{x_i},\ldots x_n\}= X$$ i.e., a set without $x_i$ in it that's an open cover of X.

Why couldn't we repeat this until the set is empty and then we get $$\bigcup \emptyset = X$$ which is obviously a contradiction.

I obviously have some deep and fundamental misunderstanding of what open covers, subcovers, and compact spaces are, so I'd appreciate any insight.

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A "subcover" does not have to be a proper subcover. That is, the finite subcover of $\{x_1,\dots,x_n\}$ which is guaranteed to exist by compactness could just be $\{x_1,\dots,x_n\}$ itself, rather than some proper subset. You are correct that if you knew every open cover had a finite proper subcover, you could repeatedly get smaller and smaller subcovers to conclude the empty set is a cover and so the space would have to be empty. (Or more simply, you could reach that conclusion by just taking a proper subcover once of the specific open cover $\{X\}$, whose only proper subset is the empty set.)

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Your error lies in the sentence “Thus, there exists some element $x_i$ s.t. $\bigcup\{x_1, \ldots, \widehat{x_i},\ldots x_n\}= X$”. That's not true. Since $B=\{x_1,x_2,\ldots,x_n\}$ is an open cover of $X$, it must have a finite subcover $B'$ of $X$, yes. But $B$ is already finite and therefore nothing prevents $B'$ from being $B$ itself.

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  • $\begingroup$ Ohhh makes sense. Unsure why I assumed it must be a proper subset and not just any subset. Thanks ! $\endgroup$
    – user975734
    Oct 3, 2021 at 20:47
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    $\begingroup$ @Obamafish given an open cover you must find finitely many of the sets in that cover that still cover. If the cover we started with was finite to start with then there is nothing to do. $\endgroup$ Oct 3, 2021 at 21:30
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    $\begingroup$ And my this upvote gives you > 362000 rep. Good luck for future! $\endgroup$
    – user876009
    Oct 4, 2021 at 5:38
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    $\begingroup$ @JitendraSingh Thanks! $\endgroup$ Oct 4, 2021 at 5:40

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