examples of locally compact and not locally compact spaces
Examples of locally compact spaces include:
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The Euclidean spaces ℝn 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 ℝ2 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 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 ℚ with the standard topology inherited from ℝ: each of its compact subsets has empty interior.
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All infinite-dimensional normed vector spaces
: a normed vector space is finite-dimensional if and only if its closed unit ball
is compact.
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The subset X={(0,0)}∪{(x,y)∣x>0} of ℝ2: no compact subset of X contains a neighborhood of (0,0).
Title | examples of locally compact and not locally compact spaces |
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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 |