View source for Aleksandrov compactification

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''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 [[#References|[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====
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<TR><TD valign="top">[1]</TD> <TD valign="top">  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}}</TD></TR>
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====Comments====
The Aleksandrov compactification is also called the one-point compactification.

====References====
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Dugundji,   "Topology" , Allyn &amp; Bacon  (1966)  (Theorem 8.4)  {{ZBL|0144.21501}}</TD></TR>
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