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. 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 Andrey Nikolayevich 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. Some authors add the assumption that the starting space X be Tychonoff (or even locally compact Hausdorff), for the following reasons: The Stone–Čech construction can be performed for more general spaces X, but the map X → βX need not be a homeomorphism to the image of X (and sometimes is not even injective). The extension property makes β a functor from Top (the category of topological spaces) to CHaus (the category of compact Hausdorff spaces). If we let U be the inclusion functor from CHaus into Top, maps from βX to K (for K in CHaus) correspond bijectively to maps from X to UK (by considering their restriction to X and using the universal property of βX). i.e. Hom(βX, K) ≅ Hom(X, UK), which means that β is left adjoint to U. This implies that CHaus is a reflective subcategory of Top with reflector β. One attempt to construct the Stone–Čech compactification of X is to take the closure of the image of X in