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 $K$ 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 homeomorphismcommutes 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

单词	PrecompactSet
释义	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 $K$ 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 homeomorphismcommutes 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
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