One-Point Compactification

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 in Y, X=Y\{y} and x in 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.

The continuous functions on Y may be of importance. Their restriction to X are loosely the continuous functions on X with a limit at infinity.

See also


This entry contributed by Allan Cortzen

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Cortzen, Allan. "One-Point Compactification." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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