Namespaces
Variants
Actions

Aleksandrov compactification

From Encyclopedia of Mathematics
Jump to: navigation, search

2010 Mathematics Subject Classification: Primary: 54D35 [MSN][ZBL]

Aleksandrov compact extension

The unique Hausdorff compactification $\alpha X$ of a locally compact, non-compact, Hausdorff space $X$, obtained by adding a single point $\infty$ to $X$. An arbitrary neighbourhood of the point $\infty$ must then have the form $\{\infty\} \cup (X \setminus F)$, where $F$ is a compact set in $X$. The Aleksandrov compactification $\alpha X$ is the smallest element in the set $B(X)$ of all compactifications of $X$. A smallest element in the set $B(X)$ exists only for a locally compact space $B(X)$ and must coincide with $\alpha X$.

The Aleksandrov compactification was defined by P.S. Aleksandrov [1] and plays an important role in topology. Thus, the Aleksandrov compactification $\alpha\mathbf{R}^n$ of the $n$-dimensional Euclidean space is identical with the $n$-dimensional sphere; the Aleksandrov compactification $\alpha\mathbf{N}$ 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 $\mathbf{R}P^2$. There are pathological situations connected with the Aleksandrov compactification, e.g. there exists a perfectly-normal, locally compact and countably-compact space $X$ whose Aleksandrov compactification has the dimensions $\dim\alpha X < \dim X$ and $\mathrm{Ind}\,\alpha X < \mathrm{Ind}\,X$.

References

[1] P.S. [P.S. Aleksandrov] Aleksandroff, "Ueber die Metrisation der im Kleinen kompakten topologischen Räumen" Math. Ann. , 92 (1924) pp. 294–301 (in German) Zbl 50.0128.04


Comments

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

References

[a1] J. Dugundji, "Topology" , Allyn & Bacon (1966) (Theorem 8.4) Zbl 0144.21501
How to Cite This Entry:
Aleksandrov compactification. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Aleksandrov_compactification&oldid=42716
This article was adapted from an original article by V.V. Fedorchuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article