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I am brand new to compactification and have a couple basic questions tied to it. I read that the one-point compactification of $\mathbb{R}$, $\mathbb{R}\cup\{\infty\}$ is compact, but struggle to see why.

My thoughts: Let's take an arbitrary open cover of $\mathbb{R}\cup\{\infty\}$, call it $(U_\alpha)_{\alpha\in A}$. Since $(U_\alpha)_{\alpha\in A}$ covers the whole space, there exists $\alpha$ such that $\infty\in U_\alpha$. Therefore, the complement of $U_\alpha$ is compact. This gives us that $U_\alpha^c$ has a finite subcover. If we add back $U_\alpha$ to this finite subcover, we have a finite cover of $\mathbb{R}\cup\{\infty\}$. However, I am concerned that this finite cover is not necessarily a subcover of our original open cover.

Am I correct in my concern? Is there some other piece I am missing?

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It is a subcover: $U_\alpha$ belongs to the original cover and the subcover that covers $U_\alpha^C$ is a subcover of the original cover. Hence, every set in the achieved subcover is an element of the original cover.

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