examples of locally compact and not locally compact spaces
Examples of locally compact spaces include:

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The Euclidean spaces ${\mathbb{R}}^{n}$ with the standard topology: their local compactness follows from the HeineBorel theorem. The complex plane (http://planetmath.org/Complex) $\u2102$ carries the same topology^{} as ${\mathbb{R}}^{2}$ and is therefore also locally compact.

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All topological manifolds^{} are locally compact since locally they look like Euclidean space.

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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 $X$).

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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.

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

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The long line is a locally compact topological space.

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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:

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

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All infinitedimensional normed vector spaces^{}: a normed vector space is finitedimensional if and only if its closed unit ball^{} is compact.

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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  20130322 12:48:36 
Last modified on  20130322 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 