One-point compactification

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In mathematics, the one-point compactification (or Alexandroff compactification) is a compactification of a topological space X, found by P. S. Alexandroff (1924).

The one-point compactification αX of X has as underlying set the disjoint union of X and an extra point, usually written as ∞. The open sets not containing ∞ are the open sets of X. The open sets containing ∞ are the complements of closed compact subsets of X.

The one-point compactification is compact and contains X as an open subspace. It is a Hausdorff space if and only if X is locally compact Hausdorff.

[edit] See also

• Stone–Čech compactification
• Wallman compactification

[edit] References

• Fedorchuk, V.V. (2001), “Aleksandrov compactification”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104 
• Alexandroff, P.S. (1924), “Über die Metrisation der im Kleinen kompakten topologischen Räume”, Math. Ann. 92: 294–301, DOI 10.1007/BF01448011