6. Compactness and Compactification

Compactness and Compactification

Definition and Properties of Compact Spaces

Compact Space: A topological space \( X \) is said to be compact if every open cover of \( X \) has a finite subcover. Equivalently, \( X \) is compact if every open cover of \( X \) can be reduced to a finite subcover.

Properties of Compact Spaces:

• Every closed subset of a compact space is compact.
• A continuous image of a compact space is compact.
• Compactness is a topological property, meaning that if \( X \) and \( Y \) are topologically equivalent, then \( X \) is compact if and only if \( Y \) is compact.

Compactification of a Topological Space

Compactification: Compactification of a topological space \( X \) is the process of adding points to \( X \) in such a way that the resulting space is compact. The compactification of \( X \) typically preserves certain properties of \( X \) while making it compact.

One example of a compactification is the one-point compactification, where an extra point is added to \( X \) to make it compact.

Examples of Compact Spaces

Finite Space: Any finite topological space is compact. This is because every open cover of a finite space can be trivially reduced to a finite subcover.

Cantor Set: The Cantor set is a compact subset of the real line \( \mathbb{R} \). It is constructed by removing open intervals from the unit interval \( [0, 1] \) in a specific manner, resulting in a set with zero measure and uncountably many points.