Stone-Čech compactification

Stone-Čech compactification is a technique for embedding a Tychonoff topological space in a compact Hausdorff space.

Let $X$ be a Tychonoff space and let $C$ be the space of all continuous functions from $X$ to the closed interval $[0,1]$. To each element $x\in X$, we may associate the evaluation functional $e_x:C\to [0,1]$ defined by $e_x(f)=f(x)$. In this way, $X$ may be identified with a set of functionals.

The space ${[0,1]}^C$ of all functionals from $C$ to $[0,1]$ may be endowed with the Tychonoff product topology. Tychonoff’s theorem asserts that, in this topology, ${[0,1]}^C$ is a compact Hausdorff space. The closure in this topology of the subset of ${[0,1]}^C$ which was identified with $X$ via evaluation functionals is $\beta X$, the Stone-Čech compactification of $X$.Being a closed subset of a compact Hausdorff space, $\beta X$ is itself a compact Hausdorff space.

This construction has the wonderful property that, for any compact Hausdorff space $Y$, every continuous function $f:X\to Y$ may be extended to a unique continuous function $\beta f:\beta X\to Y$.