Aleksandrov compactification

Aleksandrov compact extension

The unique Hausdorff compactification of a locally compact, non-compact, Hausdorff space , obtained by adding a single point to . An arbitrary neighbourhood of the point must then have the form , where is some compactum in . The Aleksandrov compactification is the smallest element in the set of all compactifications of . A smallest element in the set exists only for a locally compact space and must coincide with .

The Aleksandrov compactification was defined by P.S. Aleksandrov [1] and plays an important role in topology. Thus, the Aleksandrov compactification of the -dimensional Euclidean space is identical with the -dimensional sphere; the Aleksandrov compactification of the set of natural numbers is homeomorphic to the space of a convergent sequence together with the limit point; the Aleksandrov compactification of the "open" Möbius strip coincides with the real projective plane . There are pathological situations connected with the Aleksandrov compactification, e.g. there exists a perfectly-normal, locally compact and countably-compact space whose Aleksandrov compactification has the dimensions and .

References

[1]	P.S. [P.S. Aleksandrov] Aleksandroff, "Ueber die Metrisation der im Kleinen kompakten topologischen Räumen" Math. Ann. , 92 (1924) pp. 294–301

Comments

The Aleksandrov compactification is also called the one-point compactification.

References

[a1]	J. Dugundji, "Topology" , Allyn & Bacon (1966) (Theorem 8.4)