# Stone-Čech compactification

Stone-Čech compactification is a technique for embedding^{} a Tychonoff topological space in a compact^{} Hausdorff space.

Let $X$ be a Tychonoff space and let $C$ be the space of all continuous functions^{} from $X$ to the closed interval^{} $[0,1]$. To each element $x\in X$, we may associate the evaluation functional^{} ${e}_{x}:C\to [0,1]$ defined by ${e}_{x}(f)=f(x)$. In this way, $X$ may be identified with a set of functionals.

The space ${[0,1]}^{C}$ of *all* functionals from $C$ to $[0,1]$ may be endowed with the Tychonoff product topology. Tychonoff’s theorem^{} asserts that, in this topology^{}, ${[0,1]}^{C}$ is a compact Hausdorff space. The closure^{} in this topology of the subset of ${[0,1]}^{C}$ which was identified with $X$ via evaluation functionals is $\beta X$, the Stone-Čech compactification of $X$.
Being a closed subset of a compact Hausdorff space, $\beta X$ is itself a compact Hausdorff space.

This construction has the wonderful property that, for any compact Hausdorff space $Y$, every continuous function $f:X\to Y$ may be extended to a *unique* continuous function $\beta f:\beta X\to Y$.

Title | Stone-Čech compactification |
---|---|

Canonical name | StonevCechCompactification |

Date of creation | 2013-03-22 14:37:38 |

Last modified on | 2013-03-22 14:37:38 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 10 |

Author | rspuzio (6075) |

Entry type | Definition |

Classification | msc 54D30 |