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''Aleksandrov compact extension''
 
''Aleksandrov compact extension''
  
The unique Hausdorff [[Compactification|compactification]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011260/a0112601.png" /> of a locally compact, non-compact, Hausdorff space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011260/a0112602.png" />, obtained by adding a single point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011260/a0112603.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011260/a0112604.png" />. An arbitrary neighbourhood of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011260/a0112605.png" /> must then have the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011260/a0112606.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011260/a0112607.png" /> is some compactum in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011260/a0112608.png" />. The Aleksandrov compactification <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011260/a0112609.png" /> is the smallest element in the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011260/a01126010.png" /> of all compactifications of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011260/a01126011.png" />. A smallest element in the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011260/a01126012.png" /> exists only for a locally compact space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011260/a01126013.png" /> and must coincide with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011260/a01126014.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011260/a01126012.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011260/a01126015.png" /> of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011260/a01126016.png" />-dimensional Euclidean space is identical with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011260/a01126017.png" />-dimensional sphere; the Aleksandrov compactification <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011260/a01126018.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011260/a01126019.png" />. There are pathological situations connected with the Aleksandrov compactification, e.g. there exists a perfectly-normal, locally compact and countably-compact space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011260/a01126020.png" /> whose Aleksandrov compactification has the dimensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011260/a01126021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011260/a01126022.png" />.
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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====
 
====References====
<table><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</TD></TR></table>
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<table>
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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</TD></TR>
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</table>
  
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Dugundji,  "Topology" , Allyn &amp; Bacon  (1966)  (Theorem 8.4)</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Dugundji,  "Topology" , Allyn &amp; Bacon  (1966)  (Theorem 8.4)</TD></TR>
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</table>
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Revision as of 20:31, 10 January 2018

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 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


Comments

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

References

[a1] J. Dugundji, "Topology" , Allyn & Bacon (1966) (Theorem 8.4)
How to Cite This Entry:
Aleksandrov compactification. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Aleksandrov_compactification&oldid=16469
This article was adapted from an original article by V.V. Fedorchuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article