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.