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单词 PrecompactSet
释义

precompact set


Definition 1.

A subset in a topological spaceMathworldPlanetmath isprecompact if its closurePlanetmathPlanetmath is compactPlanetmathPlanetmath [1].

For metric spaces, we have the following theorem due to HausdorffPlanetmathPlanetmath[2].

Theorem Suppose K is a set in a complete metric space X.Then K relatively compact if and only if for any ε>0there is a finite ε-net (http://planetmath.org/VarepsilonNet) for K.

Examples

  1. 1.

    In n every point has a precompact neighborhoodMathworldPlanetmathPlanetmath.

  2. 2.

    On a manifold, every point has a precompact neighborhood.This follows from the previous example, since a homeomorphismPlanetmathPlanetmathcommutes with the closure operator, and since the continuousMathworldPlanetmath imageof a compact set is compact.

Notes

A synonym is relatively compact [2, 3].

Some authors (notably Bourbaki see [4]) use precompact differently - as a synonym for totally boundedPlanetmathPlanetmath (http://planetmath.org/TotallyBounded) (in the generality of topological groupsMathworldPlanetmath). “Relatively compact” is then used to mean “precompact ”as it is defined here

References

  • 1 J.M. Lee, Introduction to Smooth Manifolds,Graduate Texts in Mathematics series, 218,Springer-Verlag, 2002.
  • 2 R. Cristescu, Topological vector spacesMathworldPlanetmath,Noordhoff International Publishing, 1977.
  • 3 E. Kreyszig,Introductory Functional AnalysisMathworldPlanetmath With Applications,John Wiley & Sons, 1978.
  • 4 N. Bourbaki, Topological Vector Spaces Springer-Verlag, 1981
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