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If $X$ is a locally compact Hausdorff space, we can do the one-point compactification $Y$ of it. In the proof, we use the locally compactness to prove $Y$ is Hausdorff. However, I don't see anywhere I use it to prove the compactness of $Y$ except we have at least one compact set in $X$. So I am wondering if we can do one-point compactification for any space X having one compact set?

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