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 $\epsilon >0$ there is a finite $\epsilon $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
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, SpringerVerlag, 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 SpringerVerlag, 1981
Title  precompact set 

Canonical name  PrecompactSet 
Date of creation  20130322 14:39:59 
Last modified on  20130322 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 