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

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 William Elliot Posts: 2,594 Registered: 1/8/12
Re: Stone Cech
Posted: Mar 17, 2013 4:28 AM

On Fri, 15 Mar 2013, David C. Ullrich wrote:
> 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?

No. All I get is that there's a closed continuous surjection h
from Y onto X for which h|g(S) is injective.

> >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.
>

I didn't find any elevator nor even an esclator.
Are their stairs anywhere?

What's is apparent that if h,k in C(Y,X) and
f = hg, f = kg, then h = k.

> >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.
>

For latter consideration.

Perhaps you could illustrate with the five different
one to four point point compactifications of two open
end line segements.

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