View source for Stone-Čech compactification

The largest [[Compactification|compactification]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090340/s0903401.png" /> of a [[Completely-regular space|completely-regular space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090340/s0903402.png" />. Constructed by E. Čech [[#References|[1]]] and M.H. Stone [[#References|[2]]].

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090340/s0903403.png" /> be the set of all continuous functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090340/s0903404.png" />. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090340/s0903405.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090340/s0903406.png" />, is a homeomorphism onto its own image. Then, by definition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090340/s0903407.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090340/s0903408.png" /> is the operation of closure). For any compactification <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090340/s0903409.png" /> there exists a continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090340/s09034010.png" /> that is the identity on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090340/s09034011.png" />, a fact expressed by the word  "largest" .

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