precompact set

Definition 1.

A subset in a topological spaceMathworldPlanetmath is precompact 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>0 there is a finite ε-net ( for K.


  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 homeomorphismPlanetmathPlanetmath commutes with the closure operator, and since the continuousMathworldPlanetmath 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 boundedPlanetmathPlanetmath ( (in the generality of topological groupsMathworldPlanetmath). “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 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
Title precompact set
Canonical name PrecompactSet
Date of creation 2013-03-22 14:39:59
Last modified on 2013-03-22 14:39:59
Owner matte (1858)
Last modified by matte (1858)
Numerical id 12
Author matte (1858)
Entry type Definition
Classification msc 54D45
Synonym precompact
Synonym relatively compact