Namespaces
Variants
Actions

Difference between revisions of "Locally compact space"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(TeX)
Line 1: Line 1:
A topological space at every point of which there is a neighbourhood with compact closure. A locally compact Hausdorff space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060320/l0603201.png" /> is a [[Completely-regular space|completely-regular space]]. The partially ordered set of all its Hausdorff compactifications (cf. [[Compactification|Compactification]]) is a complete lattice. Its minimal element is the [[Aleksandrov compactification|Aleksandrov compactification]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060320/l0603202.png" />. The class of locally compact Hausdorff spaces coincides with the class of open subsets of Hausdorff compacta. For a locally compact Hausdorff space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060320/l0603203.png" /> its remainder <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060320/l0603204.png" /> in any Hausdorff compactification <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060320/l0603205.png" /> is a Hausdorff compactum. Every connected paracompact locally compact space is the sum of countably many compact subsets.
+
{{TEX|done}}
 +
A topological space at every point of which there is a neighbourhood with compact closure. A locally compact Hausdorff space $X$ is a [[Completely-regular space|completely-regular space]]. The partially ordered set of all its Hausdorff compactifications (cf. [[Compactification|Compactification]]) is a complete lattice. Its minimal element is the [[Aleksandrov compactification|Aleksandrov compactification]] $\alpha X$. The class of locally compact Hausdorff spaces coincides with the class of open subsets of Hausdorff compacta. For a locally compact Hausdorff space $X$ its remainder $bX\setminus X$ in any Hausdorff compactification $bX$ is a Hausdorff compactum. Every connected paracompact locally compact space is the sum of countably many compact subsets.
  
The most important example of a locally compact space is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060320/l0603206.png" />-dimensional Euclidean space. A topological Hausdorff [[Vector space|vector space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060320/l0603207.png" /> (not reducing to the zero element) over a complete non-discretely normed division ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060320/l0603208.png" /> is locally compact if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060320/l0603209.png" /> is locally compact and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060320/l06032010.png" /> is finite-dimensional over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060320/l06032011.png" />.
+
The most important example of a locally compact space is $n$-dimensional Euclidean space. A topological Hausdorff [[Vector space|vector space]] $E$ (not reducing to the zero element) over a complete non-discretely normed division ring $k$ is locally compact if and only if $k$ is locally compact and $E$ is finite-dimensional over $k$.
  
  
  
 
====Comments====
 
====Comments====
A product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060320/l06032012.png" /> of topological spaces is locally compact if and only if each separate coordinate space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060320/l06032013.png" /> is locally compact and all but finitely many are compact.
+
A product $\prod_\alpha X_\alpha$ of topological spaces is locally compact if and only if each separate coordinate space $X_\alpha$ is locally compact and all but finitely many are compact.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.L. Kelley,  "General topology" , v. Nostrand  (1955)  pp. 146–147</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.L. Kelley,  "General topology" , v. Nostrand  (1955)  pp. 146–147</TD></TR></table>

Revision as of 11:51, 6 July 2014

A topological space at every point of which there is a neighbourhood with compact closure. A locally compact Hausdorff space $X$ is a completely-regular space. The partially ordered set of all its Hausdorff compactifications (cf. Compactification) is a complete lattice. Its minimal element is the Aleksandrov compactification $\alpha X$. The class of locally compact Hausdorff spaces coincides with the class of open subsets of Hausdorff compacta. For a locally compact Hausdorff space $X$ its remainder $bX\setminus X$ in any Hausdorff compactification $bX$ is a Hausdorff compactum. Every connected paracompact locally compact space is the sum of countably many compact subsets.

The most important example of a locally compact space is $n$-dimensional Euclidean space. A topological Hausdorff vector space $E$ (not reducing to the zero element) over a complete non-discretely normed division ring $k$ is locally compact if and only if $k$ is locally compact and $E$ is finite-dimensional over $k$.


Comments

A product $\prod_\alpha X_\alpha$ of topological spaces is locally compact if and only if each separate coordinate space $X_\alpha$ is locally compact and all but finitely many are compact.

References

[a1] J.L. Kelley, "General topology" , v. Nostrand (1955) pp. 146–147
How to Cite This Entry:
Locally compact space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Locally_compact_space&oldid=11621
This article was adapted from an original article by V.V. Fedorchuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article