Review of Compactness

Topological space in which from every open cover of the space, a finite cover can be extracted.

In the study of topology, compactness is a fundamental concept that plays a crucial role in many areas of mathematics. This article provides a comprehensive review of compactness, its properties, and its significance in metric spaces.

Definition of Compactness

In topology, a set is said to be compact if every open cover of the set has a finite subcover. In simpler terms, no matter how you try to cover the set with open sets, you can always find a finite number of these open sets that still cover the entire set.

Properties of Compact Sets

Compact sets have several important properties that make them particularly interesting in the study of topology:

1. Finite Intersection Property: If a family of compact sets has the property that the intersection of every finite subcollection is nonempty, then the intersection of the entire collection is also nonempty.

2. Compact subsets of Hausdorff spaces are closed: In a Hausdorff space (a space where distinct points have disjoint neighborhoods), every compact subset is closed.

3. Image of compact sets: If a function is continuous and its domain is a compact set, then its image is also compact.

Compactness in Metric Spaces

In metric spaces, compactness has an equivalent definition: a set in a metric space is compact if and only if it is complete (all Cauchy sequences converge) and totally bounded (for any given distance, the set can be covered by finitely many balls of that radius).

Sequential Compactness and Limit Point Compactness

Sequential compactness and limit point compactness are two other forms of compactness that are often easier to work with in practice:

• A space is sequentially compact if every sequence has a subsequence that converges to a point in the space.

• A space is limit point compact if every infinite subset has a limit point in the space.

Heine-Borel Theorem

The Heine-Borel theorem is a fundamental result in real analysis that characterizes compact subsets of Euclidean space. The theorem states that a subset of Euclidean space is compact if and only if it is closed and bounded.

In conclusion, compactness is a central concept in topology with far-reaching implications in various areas of mathematics. Understanding compactness and its properties is essential for further study in topology and related fields.