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

 Messages: [ Previous | Next ]
 fom Posts: 1,968 Registered: 12/4/12
Re: Stone Cech
Posted: Mar 17, 2013 8:11 PM

On 3/17/2013 8:35 AM, David C. Ullrich wrote:
> On Sun, 17 Mar 2013 01:28:41 -0700, William Elliot <marsh@panix.com>
> wrote:
>

>> 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.

>
> I didn't read the question carefully, sorry. Sort of assumed
> it was what would seem like a sensible question regarding
> the universal property of the S-C compactification.
>
> The answer to the actual question is no, X does not embed in Y,
> or at least "surely not - the defining property of the S-C
> compactification simply has nothing to do with spaces
> embedding in Y".
>
> What's true is that X is a _quotient space_ of Y.
>

>>
>>>> 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?

>
> whether another compactification sharing the same
> universal property as the S-C compactification must
> be homeomorphic to the S-C compactification (the
> answer to _that_ is yes, and the proof starts as
> I suggested).
>
> There's simply no reason to think that the answer to
> the question you ask is yes, unless possibly it's yes
> vacuously - I can't imagine an example of (g,Y) that
> has the property you assume here.
>
> In particular, when you ask "If (g,Y) has property P,
> must (g,Y) be a S-C compactification of S?" I sort of
> assumed that property P must be a property that
> the S-C compactification actually _satisfies_.
> That's simply not so here - it's not true that every
> compactification of S embeds in the S-C compactification.
>

So, how does your last statement
reconcile with the text I quoted
from Munkres that describes Stone-Cech
compactification as maximal "in some
sense".

I realize there is a lot in the context
of definitions for terms. So I am
curious how you mean this.

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