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