This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
Unformatted text preview: 2nd April 2005 Munkres Â§ 38 Ex. 38.4. Let X â†’ Î²X be the Stoneâ€“ Ë‡ Cech compactification and X â†’ cX an arbitrary com- pactification of the completely regular space X . By the universal property of the Stoneâ€“ Ë‡ Cech compactification, the map X â†’ cX extends uniquely X / ! B B B B B B B B cX Î²X = z z z z z z z z to a continuous map Î²X â†’ cX . Any continuous map of a compact space to a Hausdorff space is closed. In particular, Î²X â†’ cX is closed. It is also surjective for it has a dense image since X â†’ cX has a dense image. Thus Î²X â†’ cX is a closed quotient map. T Ex. 38.5. (a) . For any Îµ > 0 there exists an Î± âˆˆ S Î© = [0 , Î©) such that | f ( Î² )- f ( Î± ) | < Îµ for all Î² > Î± . For if no such element existed we could find an increasing sequence of elements Î³ n âˆˆ (0 , Î©) such that | f ( Î³ n )- f ( Î³ n- 1 ) | â‰¥ Îµ for all n . But any increasing sequence in (0 , Î©) converges to its least upper bound whereas the image sequence f ( Î³ n ) âˆˆ R does not converge; this contradicts continuity of the function f : (0 , Î©) â†’ R . So in particular, there exist elements Î± n such that | f ( Î² )- f ( Î± n ) | < 1 /n for all Î² > Î± n . Let Î± be an upper bound for these elements. Then f is constant on ( Î±, Î©). (b) . Since any real function on (0 , Î©) is eventually constant, any real function, in particular any bounded real function, on (0 , Î©) extends to the one-point-compactification (0 , Î©]. But the Stone- Ë‡ Cech compactification is characterized by this property [Thm 38.5] so (0 , Î©] = Î² (0 , Î©). (c) . Use that any compactification of (0 , Î©) is a quotient of (0 , Î©] [Ex 38.4]....
View Full Document
- Fall '08
- Topology, Topological space, General topology, Hausdorff space, Î²x, completely regular space