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) |
Aleksandrov compactification. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Aleksandrov_compactification&oldid=42709