
Re: Compactification
Posted:
Dec 14, 2012 7:04 PM


On Dec 14, 3:14 pm, David Hartley <m...@privacy.net> wrote: > > IS it possible to have two homeomorphic compactifications of X such that > no homeomorphism f:Y > Z has f(X) = X ? or such that Y\X and Z\X are > not homeomorphic ?
Yes. Here, if I'm not mistaken, is an example of a compact space K (a subspace of the complex plane) with two dense subspaces X and X' such that X is homeomorphic to X' while K\X is not homeomorphic to K\X'.
K = {z: z+2 <= 1 OR z <= 1 OR z2 <= 1}.
A = {z: z+2 < 1}, B = {z: z < 1}, C = {z: z2 < 1}.
A_0 is a countable dense subset of A, B_0 is a countable dense subset of B, C_0 is a countable dense subset of C.
X is the union of A, B_0, and C_0; X' is the union of A_0, B, and C_0.

