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Math Forum » Discussions » sci.math.* » sci.math.independent

Topic: Compactification
Replies: 15   Last Post: Mar 17, 2013 6:11 AM

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Butch Malahide

Posts: 885
Registered: 6/29/05
Re: Compactification
Posted: Dec 14, 2012 7:04 PM
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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 |z-2| <= 1}.

A = {z: |z+2| < 1}, B = {z: |z| < 1}, C = {z: |z-2| < 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.

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