Let be a Tychonoff space and let be the space of all continuous functions from to the closed interval . To each element , we may associate the evaluation functional defined by . In this way, may be identified with a set of functionals.
The space of all functionals from to may be endowed with the Tychonoff product topology. Tychonoff’s theorem asserts that, in this topology, is a compact Hausdorff space. The closure in this topology of the subset of which was identified with via evaluation functionals is , the Stone-Čech compactification of . Being a closed subset of a compact Hausdorff space, is itself a compact Hausdorff space.
This construction has the wonderful property that, for any compact Hausdorff space , every continuous function may be extended to a unique continuous function .
|Date of creation||2013-03-22 14:37:38|
|Last modified on||2013-03-22 14:37:38|
|Last modified by||rspuzio (6075)|