On Mon, 18 Mar 2013 01:50:39 -0700, William Elliot <firstname.lastname@example.org> wrote:
>Let (f,X) and (y,Y) be compactifications of S. >Assume h in C(Y,X) and f = hg. > >Thue h is a continuous surjection and when Y is Hausdorff >a closed quotient map. > >k = h|g(S):g(S) -> f(S) is a continuous bijection. >It it a homeomorphism? If so, what's a proof like?
I doubt that it follows that k is a homeomorphism, although I'm not going to try to give a counterexample.
Assuming it doesn't follow, then k being a homeomorphism "should" be part of the _definition_ of the partial order on the class of compactifications that we're struggling to define here.