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 is nonempty do not give an explicit description of any particular point in .
The Stone–Čech compactification was found by Marshall Stone (1937) and Eduard Čech (1937).
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 : X → K, where K is a compact Hausdorff space, lifts uniquely to a continuous map βf : βX → K.
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