# Stone-Čech compactification

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