# Aleksandrov compactification

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 |

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Aleksandrov compactification.

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