Examples of locally compact spaces include:

• The Euclidean spaces $\Bbb{R}^n$ with the standard topology: their local compactness follows from the Heine-Borel theorem. The complex plane $\Bbb{C}$ carries the same topology as $\Bbb{R}^2$ 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 of two closed subsets of $X$ (equivalently: the intersection of an open and a closed subset of $X$ ).
• The space of $p$ -adic rationals is homeomorphic to the Cantor set minus one point, and since the Cantor set is compact as a closed bounded subset of $\Bbb{R}$ , 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 $\Bbb{Q}$ with the standard topology inherited from $\Bbb{R}$ : 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 $\Bbb{R}^2$ : no compact subset of $X$ contains a neighborhood of $(0,0)$ .