Stone-Čech Compactification

The (up to unique homeomorphism) largest compactification βX of a completely regular (Tychonoff) space X characterized by the property that every bounded continuous real-valued function on X extends uniquely to a continuous function on βX.

For a completely regular (Tychonoff) space X, the Stone‑Čech compactification βX is a compact Hausdorff space equipped with a dense embedding i: X → βX characterized by the universal property that every continuous map from X to any compact Hausdorff space K extends uniquely to a continuous map βX → K.