Difference between revisions of "Stone-Čech compactification"
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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 [[ | + | 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==== | ====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> | + | <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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====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> | + | <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> | ||
+ | |||
+ | {{TEX|done}} |
Latest revision as of 10:44, 19 October 2016
2020 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) |
Stone-Čech compactification. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stone-%C4%8Cech_compactification&oldid=14277