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examples of locally compact and not locally compact spaces


Examples of locally compact spaces include:

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

  • All topological manifoldsMathworldPlanetmathPlanetmath 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 spacePlanetmathPlanetmath 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 homeomorphicMathworldPlanetmath to the Cantor set minus one point, and since the Cantor set is compactPlanetmathPlanetmath as a closed boundedPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath subset of , 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 setMathworldPlanetmath 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 HausdorffPlanetmathPlanetmath.

Examples of spaces which are not locally compact include:

  • The rational numbers with the standard topology inherited from : each of its compact subsets has empty interior.

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

  • The subset X={(0,0)}{(x,y)x>0} of 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