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precompact set
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(Definition)
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For metric spaces, we have the following theorem due to Hausdorff [2].
Theorem Suppose  is a set in a complete metric space  . Then  relatively compact if and only if for any
 there is a finite
 -net for  .
- In
 every point has a precompact neighborhood.
- On a manifold, every point has a precompact neighborhood. This follows from the previous example, since a homeomorphism commutes with the closure operator, and since the continuous image of a compact set is compact.
A synonym is relatively compact [2,3].
Some authors (notably Bourbaki see [4]) use precompact differently - as a synonym for totally bounded (in the generality of topological groups). “Relatively compact” is then used to mean “precompact ”as it is defined here
- 1
- J.M. Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics series, 218, Springer-Verlag, 2002.
- 2
- R. Cristescu, Topological vector spaces, Noordhoff International Publishing, 1977.
- 3
- E. Kreyszig, Introductory Functional Analysis With Applications, John Wiley & Sons, 1978.
- 4
- N. Bourbaki, Topological Vector Spaces Springer-Verlag, 1981
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"precompact set" is owned by matte. [ full author list (3) ]
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(view preamble)
| Other names: |
precompact, relatively compact |
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Cross-references: mean, topological groups, Bourbaki, continuous image of a compact set is compact, closure operator, homeomorphism, manifold, neighborhood, point, finite, complete, Hausdorff, metric spaces, compact, closure, topological space, subset
There are 10 references to this entry.
This is version 9 of precompact set, born on 2004-10-01, modified 2006-07-13.
Object id is 6264, canonical name is PrecompactSet.
Accessed 7430 times total.
Classification:
| AMS MSC: | 54D45 (General topology :: Fairly general properties :: Local compactness, $\sigma$-compactness) |
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Pending Errata and Addenda
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