# 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. 1.

In $\mathbb{R}^{n}$ every point has a precompact neighborhood.

2. 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 PrecompactSet 2013-03-22 14:39:59 2013-03-22 14:39:59 matte (1858) matte (1858) 12 matte (1858) Definition msc 54D45 precompact relatively compact