Let $(X, \tau)$ be a topological space. The one-point compactification $X^+$ is a compact [Hausdorff space](/page/Hausdorff%20Space) if and only if $X$ is locally compact, Hausdorff, and not compact.
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When $X$ is compact, the point $\infty$ is isolated in $X^+$ (since $X$ itself is compact, so $\{\infty\} = (X \setminus X) \cup \{\infty\}$ is open), and $X^+$ is the disjoint union of $X$ and an isolated point. The construction is mainly of interest when $X$ is not compact.