View source for Stone-Čech compactification

{{MSC|54D35}}

The largest [[compactification]] $\beta X$ of a [[completely-regular space]] $X$. Constructed by E. Čech [[#References|[1]]] and M.H. Stone [[#References|[2]]].

Let $\{ f_\alpha : X \rightarrow [0,1] \}_{\alpha \in A}$ be the set of all continuous functions $X \rightarrow [0,1]$. The mapping $\phi : X \rightarrow \mathbf{R}^A$, where $\phi(X)_\alpha = f_\alpha(X)$, is a homeomorphism onto its own image. Then, by definition, $\beta X = [\phi(X)]$ (where $[ \cdot ]$ denotes the operation of [[Closure of a set|closure]]). For any compactification $b X$ there exists a continuous mapping $\beta X \rightarrow b X$ that is the identity on $X$, a fact expressed by the word  "largest" .

The Stone–Čech compactification of a [[quasi-normal space]] coincides with its [[Wallman compactification]].

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  E. Čech,   "On bicompact spaces"  ''Ann. of Math.'' , '''38'''  (1937)  pp. 823–844</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  M.H. Stone,   "Applications of the theory of Boolean rings to general topology"  ''Trans. Amer. Soc.'' , '''41'''  (1937)  pp. 375–481</TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top">  R. Engelking,   "Outline of general topology" , North-Holland  (1968)  (Translated from Polish)</TD></TR>
<TR><TD valign="top">[4]</TD> <TD valign="top">  P.S. Aleksandrov,   "Some results in the theory of topological spaces, obtained within the last twenty-five years"  ''Russian Math. Surveys'' , '''15''' :  2  (1960)  pp. 23–83  ''Uspekhi Mat. Nauk'' , '''15''' :  2  (1960)  pp. 25–95</TD></TR>
</table>



====Comments====
Instead of Stone–Čech compactification one finds about equally frequently Čech–Stone compactification in the literature.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Engelking,   "General topology" , Heldermann  (1989)</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Gillman,   M. Jerison,   "Rings of continuous functions" , Springer  (1976)</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  J.R. Porter,   R.G. Woods,   "Extensions and absolutes of Hausdorff spaces" , Springer  (1988)</TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top">  R.C. Walker,   "The Stone–Čech compactification" , Springer  (1974)</TD></TR>
</table>

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