# Difference between revisions of "Locally compact space"

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− | 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. | + | 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 space|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 $k$. | 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 $k$. |

## Latest revision as of 16:58, 15 April 2018

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=34468