precompact set

Definition 1.

A subset in a topological space is precompact if its closure is compact [1].

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

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

Examples

1.

In $\mathbb{R}^{n}$ 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 image of 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

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