Stone-Čech compactification

In the mathematical discipline of general topology, Stone–Čech compactification (or Čech–Stone 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 βX \ X is nonempty do not give an explicit description of any particular point in βX \ X.

The Stone–Čech compactification occurs implicitly in a paper by Tychonoff (1930) and was given explicitly by Marshall Stone (1937) and Eduard Čech (1937).

The Stone–Čech compactification βX is a compact Hausdorff space together with a continuous map from X that has the following universal property: any continuous map f: X → K, where K is a compact Hausdorff space, extends uniquely to a continuous map βf: βX → K.

As is usual for universal properties, this universal property, together with the fact that βX is a compact Hausdorff space containing X, characterizes βX up to homeomorphism.

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