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Stone-Čech compactification

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The largest compactification of a completely-regular space . Constructed by E. Čech [1] and M.H. Stone [2].

Let be the set of all continuous functions . The mapping , where , is a homeomorphism onto its own image. Then, by definition, (where is the operation of closure). For any compactification there exists a continuous mapping that is the identity on , 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://www.encyclopediaofmath.org/index.php?title=Stone%E2%80%93%C4%8Cech_compactification&oldid=23534