Stone-Čech compactification

Let X be a Tychonoff space and let C be the space of all continuous functionsMathworldPlanetmathPlanetmath from X to the closed intervalMathworldPlanetmath [0,1]. To each element xX, we may associate the evaluation functionalMathworldPlanetmathPlanetmathPlanetmath ex:C[0,1] defined by ex(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 theoremMathworldPlanetmath asserts that, in this topologyMathworldPlanetmathPlanetmath, [0,1]C is a compact Hausdorff space. The closureMathworldPlanetmathPlanetmath in this topology of the subset of [0,1]C which was identified with X via evaluation functionals is βX, the Stone-Čech compactification of X. Being a closed subset of a compact Hausdorff space, βX is itself a compact Hausdorff space.

This construction has the wonderful property that, for any compact Hausdorff space Y, every continuous function f:XY may be extended to a unique continuous function βf:βXY.

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