Compact space

Compact

Compact space

In mathematics, particularly in topology, a compact space is a type of topological space that, in a certain sense, is limited in size. The concept of compactness is a generalization of the notion of a closed and bounded subset of Euclidean space.

Definition

A topological space \( X \) is said to be compact if every open cover of \( X \) has a finite subcover. This means that for any collection of open sets whose union includes \( X \), there exists a finite number of these open sets that still cover \( X \).

Examples

• The closed interval \([0, 1]\) in the real numbers with the standard topology is a compact space.
• Any finite space is compact.
• The Cantor set is an example of a compact space.

Properties

• A compact space is always Lindelöf and paracompact.
• In a Hausdorff space, compact subsets are closed.
• The continuous image of a compact space is compact.
• A compact subset of a metric space is sequentially compact.

Compactness in Metric Spaces

In the context of metric spaces, a subset is compact if and only if it is both sequentially compact and totally bounded.

Related Concepts

• Sequential compactness
• Limit point compactness
• Local compactness
• Paracompact space
• Lindelöf space
• Tychonoff theorem

Applications

Compact spaces are fundamental in various areas of mathematics, including analysis, algebraic topology, and functional analysis. They are used in the study of continuous functions, differential equations, and measure theory.

See Also

• Topological space
• Metric space
• Hausdorff space
• Open cover
• Closed set
• Bounded set

Related Pages

• Topological space
• Metric space
• Hausdorff space
• Open cover
• Closed set
• Bounded set