Compactification

Definition

Given a topological space $X$, a compactification of $X$ is defined as a topological space $Y$ having $X$ as a subspace, such that:

• $Y$ is a compact Hausdorff space
• $X$ is a dense subset of $Y$

A topological space can possess a compactification if and only if it is a completely regular space. This is because the property of being completely regular is subspace-hereditary, and every copmact Hausdorff space is completely regular. A compactification process that works for all completely regular spaces, is the Stone-Cech compactification