Locally compact space

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