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

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 David Hartley Posts: 463 Registered: 12/13/04
Re: Compactification
Posted: Dec 15, 2012 10:46 AM

In message
Butch Malahide <fred.galvin@gmail.com> writes
>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.

Nice!

I thought it should be possible but couldn't come up with an example
myself.

--
David Hartley

Date Subject Author
12/11/12 William Elliot
12/11/12 magidin@math.berkeley.edu
12/13/12 William Elliot
12/13/12 magidin@math.berkeley.edu
12/14/12 William Elliot
12/14/12 magidin@math.berkeley.edu
12/14/12 David Hartley
12/14/12 Butch Malahide
12/15/12 David Hartley
12/12/12 Shmuel (Seymour J.) Metz
12/13/12 William Elliot
12/13/12 Shmuel (Seymour J.) Metz
12/12/12 David C. Ullrich
12/13/12 William Elliot
3/17/13 fom
3/17/13 fom