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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]]].
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{{MSC|54D35}}
  
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" .
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The largest [[compactification]] $\beta X$ of a [[completely-regular space]] $X$. Constructed by E. Čech [[#References|[1]]] and M.H. Stone [[#References|[2]]].
  
The Stone–Čech compactification of a [[Quasi-normal space|quasi-normal space]] coincides with its [[Wallman compactification|Wallman compactification]].
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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" .
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The Stone–Čech compactification of a [[quasi-normal space]] coincides with its [[Wallman compactification]].
  
 
====References====
 
====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>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  E. Čech,  "On bicompact spaces"  ''Ann. of Math.'' , '''38'''  (1937)  pp. 823–844</TD></TR>
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<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>
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<TR><TD valign="top">[3]</TD> <TD valign="top">  R. Engelking,  "Outline of general topology" , North-Holland  (1968)  (Translated from Polish)</TD></TR>
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<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>
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</table>
  
  
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====References====
 
====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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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Engelking,  "General topology" , Heldermann  (1989)</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Gillman,  M. Jerison,  "Rings of continuous functions" , Springer  (1976)</TD></TR>
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<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>
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<TR><TD valign="top">[a4]</TD> <TD valign="top">  R.C. Walker,  "The Stone–Čech compactification" , Springer  (1974)</TD></TR>
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</table>
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Latest revision as of 10:44, 19 October 2016

2010 Mathematics Subject Classification: Primary: 54D35 [MSN][ZBL]

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

[1] E. Čech, "On bicompact spaces" Ann. of Math. , 38 (1937) pp. 823–844
[2] M.H. Stone, "Applications of the theory of Boolean rings to general topology" Trans. Amer. Soc. , 41 (1937) pp. 375–481
[3] R. Engelking, "Outline of general topology" , North-Holland (1968) (Translated from Polish)
[4] 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


Comments

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

References

[a1] R. Engelking, "General topology" , Heldermann (1989)
[a2] L. Gillman, M. Jerison, "Rings of continuous functions" , Springer (1976)
[a3] J.R. Porter, R.G. Woods, "Extensions and absolutes of Hausdorff spaces" , Springer (1988)
[a4] R.C. Walker, "The Stone–Čech compactification" , Springer (1974)
How to Cite This Entry:
Stone-Čech compactification. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stone-%C4%8Cech_compactification&oldid=39436
This article was adapted from an original article by I.G. Koshevnikova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article