# Stone–Čech compactification

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In the mathematical discipline of general topology, Stone–Čech compactification is a technique for constructing a universal map from a topological space X to a compact Hausdorff space βX. The Stone–Čech compactification βX of a topological space X is the largest compact Hausdorff space "generated" by X, in the sense that any map from X to a compact Hausdorff space factors through βX (in a unique way). If X is a Tychonoff space then the map from X to its image in βX is a homeomorphism, so X can be thought of as a (dense) subspace of βX. For general topological spaces X, the map from X to βX need not be injective.

A form of the axiom of choice is required to prove that every topological space has a Stone–Čech compactification. Even for quite simple spaces X, an accessible concrete description of βX often remains elusive. In particular, proofs that $\beta \mathbb{N} \setminus \mathbb{N}$ is nonempty do not give an explicit description of any particular point in $\beta \mathbb{N} \setminus \mathbb{N}$.

The Stone–Čech compactification was found by Marshall Stone (1937) and Eduard Čech (1937).

## Contents

### Universal property and functoriality

βX is a compact Hausdorff space together with a continuous map from X and has the following universal property: any continuous map f : XK, where K is a compact Hausdorff space, lifts uniquely to a continuous map βf : βXK.