On Apr 22, 9:56 am, dullr...@sprynet.com wrote: > On Sun, 21 Apr 2013 18:55:29 -0700, William Elliot <ma...@panix.com> > wrote: > >Can a perfect compact Hausdorff space be continuously mapped onto [0,1]? > [. . .] > The answer to the question you meant to ask is also yes, it seems > to me. Say K is a perfect compact Hausdorff space. Either > K has a connected subset containing more than one point or not. > > If C is a subset of K containing p and q, p <> q, then it follows > from Tietze that there is a continuous f : K -> [0,1] with > f(p) = 0 and f(q) = 1; now f(C) must be connected, qed. > > On the other hand, if K has no connected subsets with > more than one point: It's easy to construct a continuous > map from K onto the Cantor set, qed. (Say K is the disjoint > union of the closed set A_0 and A_1. Map A_0 to the > left half of the Cantor set and A_1 to the right half. > Now A_0 is the disjoint union of the closed sets > A_00 and A_0,1... repeat countably many times > and you've defined a continuous map onto the Cantor set.)
Nice. I think you can combine the two cases into one.
Let A_0 and A_1 be disjoint perfect subsets of K. Construct a continuous f_1 : K -> [0,1] with f_1(A_0) = 0 and f_1(A_1) = 1.
Let A_00 and A01 be disjoint perfect subsets of A_0, and let A_10 and A_11 be disjoint perfect subsets of A_1. Construct a continuous f_2 : K -> [0,1] with f_2(A_00 \/ A_10) = 0 and f_2(A_01 \/ A_11) = 1.
Continuing in this way, define a continuous function f_n : K -> [0,1] for each natural n. Finally, the function