Proving a Set is Compact: Methods and Examples

Introduction

In advanced mathematics, the concept of a compact set is central to many areas including analysis, topology, and functional analysis. A compact set is a fundamental topological property that ensures well-behaved properties such as the ability to extract convergent subsequences, and it is a powerful tool in proofs and applications. This article will explore the methods to prove a set is compact, with specific examples to clarify the concepts.

1. The Heine-Borel Theorem for Euclidean Spaces

The Heine-Borel Theorem provides a straightforward method to prove compactness for subsets of Euclidean spaces, specifically in mathbb{R}^n. A subset S of mathbb{R}^n is compact if and only if it is both closed and bounded.

Definitions:

Closed Set: A set that contains all its limit points. Bounded Set: A set that can be contained within a ball of finite radius.

Example: To prove that the closed interval [a, b] in mathbb{R} is compact:

**Closed:** The interval [a, b] contains all its limit points; any point that can be approached by points in the interval. **Bounded:** The interval is contained within the finite bounds a and b.

2. The Open Cover Definition

The open cover definition is a more general method that applies to any topological space. A set S is compact if every open cover of S has a finite subcover. An open cover is a collection of open sets whose union contains S.

Steps to Prove:

Assume you have an open cover of S. Show that you can extract a finite number of these open sets that still cover S.

Example: Consider the closed interval [a, b] in mathbb{R}. To prove it is compact using the open cover definition:

Consider any open cover of [a, b]. Show that a finite number of open sets from this cover can still cover [a, b].

3. Sequential Compactness in Metric Spaces

Sequential compactness is another criterion for compactness, specifically in metric spaces. A set S is sequentially compact if every sequence in S has a subsequence that converges to a limit that is also in S.

Steps to Prove:

Take any sequence in S. Show that there exists a convergent subsequence whose limit is in S.

Example: To prove [a, b] is sequentially compact in mathbb{R} using sequential compactness:

Consider any sequence in [a, b]. Show that a subsequence of this sequence converges to a limit within [a, b].

4. Limit Point Compactness

Limit point compactness is a related concept in topological spaces. A set S is limit point compact if every infinite subset of S has at least one limit point in S.

Example:

Consider an infinite subset of [a, b]. Show that this subset has a limit point within [a, b].

Conclusion

Choosing the appropriate method to prove compactness depends on the context and the space you are working with. For metric spaces, sequential compactness is often the easiest to apply. However, the open cover definition is more general and applies to any topological space, making it a more universal approach.