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 for $K$ .

Examples

1. In $\sR^n$ 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 (in the generality of topological groups). ``Relatively compact'' is then used to mean ``precompact ''as it is defined here

Bibliography

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