examples of locally compact and not locally compact spaces

Examples of locally compact spaces include:

•

The Euclidean spaces $\mathbb{R}^{n}$ with the standard topology: their local compactness follows from the Heine-Borel theorem. The complex plane (http://planetmath.org/Complex) $\mathbb{C}$ carries the same topology as $\mathbb{R}^{2}$ and is therefore also locally compact.
•

All topological manifolds are locally compact since locally they look like Euclidean space.
•

Any closed or open subset of a locally compact space is locally compact. In fact, a subset of a locally compact Hausdorff space $X$ is locally compact if and only if it is the difference (http://planetmath.org/SetDifference) of two closed subsets of $X$ (equivalently: the intersection of an open and a closed subset of $X$ ).
•

The space of $p$ -adic rationals (http://planetmath.org/PAdicIntegers) is homeomorphic to the Cantor set minus one point, and since the Cantor set is compact as a closed bounded subset of $\mathbb{R}$ , we see that the $p$ -adic rationals are locally compact.
•

Any discrete space is locally compact, since the singletons can serve as compact neighborhoods.
•

The long line is a locally compact topological space.
•

If you take any unbounded totally ordered set and equip it with the left order topology (or right order topology), you get a locally compact space. This space, unlike all the others we have looked at, is not Hausdorff.

Examples of spaces which are not locally compact include:

•

The rational numbers $\mathbb{Q}$ with the standard topology inherited from $\mathbb{R}$ : each of its compact subsets has empty interior.
•

All infinite-dimensional normed vector spaces: a normed vector space is finite-dimensional if and only if its closed unit ball is compact.
•

The subset $X=\{(0,0)\}\cup\{(x,y)\mid x>0\}$ of $\mathbb{R}^{2}$ : no compact subset of $X$ contains a neighborhood of $(0,0)$ .

Title	examples of locally compact and not locally compact spaces
Canonical name	ExamplesOfLocallyCompactAndNotLocallyCompactSpaces
Date of creation	2013-03-22 12:48:36
Last modified on	2013-03-22 12:48:36
Owner	AxelBoldt (56)
Last modified by	AxelBoldt (56)
Numerical id	18
Author	AxelBoldt (56)
Entry type	Example
Classification	msc 54D45
Related topic	TopologicalSpace