One-point compactification

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If Y is a compact, Hausdorff space; X is a topological space such that Y-X contains one element and the closure of X equals Y; then Y is the one-point compactification of X. The One-point compactification is the minimal compactification one can perform on X. A space X has a one-point compactification if and only if it is itself locally compact and Hausdorff.