precompact set
Definition 1.
A subset in a topological space isprecompact if its closure
is compact
[1].
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 (http://planetmath.org/VarepsilonNet) for .
Examples
- 1.
In every point has a precompact neighborhood
.
- 2.
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
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 bounded (http://planetmath.org/TotallyBounded) (in the generality of topological groups
). “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 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