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Topic: Stone Cech
Replies: 49   Last Post: Mar 28, 2013 12:15 PM

 Messages: [ Previous | Next ]
 David C. Ullrich Posts: 21,553 Registered: 12/6/04
Re: Stone Cech
Posted: Mar 15, 2013 9:17 AM

On Thu, 14 Mar 2013 19:49:08 -0700, William Elliot <marsh@panix.com>
wrote:

>Let (g,Y) be a Cech Stone compactification of S.
>If (f,X) is a compactification of S, does X embed in Y?

Isn't this clear from the universal property of the S-C
compactification?

>
>If (g,Y) is a compactification of S and
>for all compactifications (f,X), X embeds in Y
>is (g,Y) a Stone Cech compactification of S?

My guess is that there is a counterexample to
exactly what you wrote here. The universal
property of bS says that any map from S
to a compact space lifts _uniquely_ to a map
from bS to the same space. If you add the
same uniqueness bit to your hypotheses on
(g,Y) then surely the answer is yes; consider
the case where (f,X) is the S-C compactifiication
to get started.

>Why in the heck is a compactification an embedding
>function and a compact space? Wouldn't be simpler
>to define a compactification of a space S, as a
>compact space into which S densely embeds?

No. For the same reasons as the last time you
asked. Or consider this: Part of the definition
of the S-C compactification is the _uniqueness_
of the lift of a map from S to any compact space.
Let's say the notion you're considering is a
WEification. A WEification could correspond
to two different compactifications in the
usual sense; now the uniqueness of the lift
can't even be defined, much less asserted.

Date Subject Author
3/14/13 William Elliot
3/14/13 fom
3/15/13 fom
3/16/13 William Elliot
3/15/13 David C. Ullrich
3/17/13 William Elliot
3/17/13 David C. Ullrich
3/17/13 fom
3/18/13 David C. Ullrich
3/18/13 fom
3/18/13 David Hartley
3/19/13 William Elliot
3/19/13 David Hartley
3/19/13 William Elliot
3/20/13 Butch Malahide
3/20/13 David C. Ullrich
3/20/13 Butch Malahide
3/20/13 Butch Malahide
3/21/13 quasi
3/21/13 quasi
3/21/13 quasi
3/21/13 quasi
3/21/13 Butch Malahide
3/21/13 quasi
3/22/13 Butch Malahide
3/22/13 Butch Malahide
3/22/13 Butch Malahide
3/22/13 quasi
3/22/13 David C. Ullrich
3/22/13 David C. Ullrich
3/22/13 Butch Malahide
3/23/13 Butch Malahide
3/23/13 David C. Ullrich
3/23/13 David C. Ullrich
3/23/13 Frederick Williams
3/23/13 David C. Ullrich
3/23/13 Frederick Williams
3/22/13 Butch Malahide
3/23/13 David C. Ullrich
3/22/13 Butch Malahide
3/23/13 quasi
3/23/13 Butch Malahide
3/23/13 Butch Malahide
3/24/13 quasi
3/24/13 Frederick Williams
3/24/13 quasi
3/25/13 Frederick Williams
3/28/13 Frederick Williams
3/25/13 quasi
3/19/13 David C. Ullrich