Editing Stone–Čech compactification (section)

{{Short description|Concept in topology}}
In the mathematical discipline of [[general topology]], '''Stone–Čech compactification''' (or '''Čech–Stone compactification'''<ref>M. Henriksen, "Rings of continuous functions in the 1950s", in ''Handbook of the History of General Topology'', edited by C. E. Aull, R. Lowen,
Springer Science & Business Media, 2013, p. 246</ref>) is a technique for constructing a [[Universal property|universal map]] from a [[topological space]] ''X'' to a [[Compact space|compact]] [[Hausdorff space]] ''&beta;X''. The Stone–Čech compactification ''βX'' of a topological  space ''X'' is the largest, most general compact Hausdorff space "generated" by ''X'', in the sense that any continuous map from ''X'' to a compact Hausdorff space [[List of mathematical jargon#factor through|factors through]] ''βX'' (in a unique way). If ''X'' is a [[Tychonoff space]] then the map from ''X'' to its [[image (mathematics)|image]] in ''βX'' is a [[homeomorphism]], so ''X'' can be thought of as a ([[Dense (topology)|dense]]) subspace of ''βX''; every other compact Hausdorff space that densely contains ''X'' is a [[Quotient space (topology)|quotient]] of ''βX''. For general topological spaces ''X'', the map from ''X'' to ''βX'' need not be [[Injective function|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 {{math|''βX'' &setminus; ''X''}} is nonempty do not give an explicit description of any particular point in {{math|''βX'' &setminus; ''X''}}.

The Stone–Čech compactification occurs implicitly in a paper by  {{harvs|txt|authorlink=Andrey Nikolayevich Tikhonov|last=Tychonoff|first=Andrey Nikolayevich|year=1930}} and was given explicitly by {{harvs|authorlink=Marshall Stone|first=Marshall|last=Stone|year=1937|txt=yes}} and {{harvs|authorlink=Eduard Čech|first=Eduard |last=Čech|year=1937|txt=yes}}.