A topological space X has a one-point compactification if and only if it is locally compact. To see a part of this, assume Y is compact, y element Y, X = Y\{y} and x element X. Let C be a compact neighborhood of x (relative to Y), not containing y. Then C is also compact relative to X, which shows X is locally compact. The point y is often called the point of infinity. A one-point compactification opens up for simplifications in definitions and proofs.
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