examples of locally compact and not locally compact spaces

Examples of locally compact spaces include:

• The Euclidean spaces ${ℝ}^n$ with the standard topology: their local compactness follows from the Heine-Borel theorem. The complex plane (http://planetmath.org/Complex) $ℂ$ carries the same topology as ${ℝ}^2$ 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 $ℝ$, 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 $ℝ$: 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 ${ℝ}^2$: no compactsubset of $X$ contains a neighborhood of $(0,0)$.