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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 17, 2013 9:35 AM

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.

Possibly you've misunderstood the universal property
of the S-C compactification. If not, you seem to be
just making up random questions, asking whether
a certain property characterizes the S-C compactification
when that property simply has nothing to do with it.

>> 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