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

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David Hartley

Posts: 441
Registered: 12/13/04
Re: Compactification
Posted: Dec 14, 2012 4:14 PM
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In message <09712de9-7a3f-4b08-b876-a465c640cf1e@googlegroups.com>,
Arturo Magidin <magidin@member.ams.org> writes
>The one-point compactification is a quotient of the 2-point
>compactification. Any map that extends to the one-point
>compactification will necessarily extend to the 2 point
>compactification.


...but not vice-versa. The only compactification to which every suitable
map extends is the Stone-Cech. Some even define it by that property (
en.wikipedia.org/wiki/Stone?C(ech_compactification ). So it is hardly
a "usual" property of compactifications.

Going back to the original question, there doesn't seem to be much point
in empathising the embedding used unless different embeddings in the
same compact space can give different compactifications. I.e. is it
possible to have two compact spaces Y and Z each containing a dense
subspace X such that Y and Z are homeomorphic but there is no
homeomorphism between them which is the identity on X.

... and that is indeed possible. For example, consider two two-point
compactifications of N. one the sum of the one-point compactifications
of the odds and the evens, the other of the one-point compactifications
of the primes and the composites.

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 ?
--
David Hartley




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