Stone-Cech Compactification

Definition

Consider a Completely Regular Space $(X,\mathcal{T})$. Let $C(X,\mathbb{R})$ denote the set of all bounded continuous functions on $X$ to $\mathbb{R}$. Then there exists a Compactification $\beta (X)$ of $X$ such that every element of $C(X,\mathbb{R})$ can be uniquely extended to an element of $C(\beta (X),\mathbb{R})$. Where the compactification $\beta (X)$ is called the Stone-Cech compactification of $X$.