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

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 David C. Ullrich Posts: 21,553 Registered: 12/6/04
Re: Stone Cech
Posted: Mar 18, 2013 11:06 AM

On Sun, 17 Mar 2013 19:11:00 -0500, fom <fomJUNK@nyms.net> wrote:

>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".

(Under suitable hypotheses; being an analyst
assuming locally compact Hausdorff is fine
with me:)

The S-C of X, let's call it bX, is maximal
in the sense that every compactification
is a _quotient space_ of bX. That's
simply a totally different thing from
saying every compactification embeds
in bX, which is not true (or if it is true,
it's true just by accident).

Consider R, the real line. The space bX
consists of R with a huge amount of
fuzzy sttuff tacked on at the ends.
The one-point compactification of R
is obtained from bX by taking all the points
other than points of R and "identifying"
them to a single point.

Otoh does the one-point compactification
of R embed in bR? I don't know for sure,
but I doubt it. And it _is_ clear that it doesn't
embed in the relevant sense:

The question is whether bR contains
something homeomorphic to S^1. I doubt
it. But it's clear that there is no S^1 in bR
that consists of R plus one more point,
which is what I mean by saying there's no
"relevant" embedding of S^1 in bR. If
there is an S^1 in bR it's just a random
circle sitting somewhere in that fuzzy
stuff, that really has nothing to do with
the fact that bR is bR.

Hmm. This gives a better answer to the OP's
question about why a compactiification is defined
as an ordered pair including an embedding, instead
of just as a topological space. If a compactification
of X is a topological space then the notion of one
compactification embedding in another includes
embeddings that are simply irrelevant to X.

Otoh it's easy to give a definition that captures
what it "really means", or should mean, for one
compactification to embed in another. Say
(g,A) is a compactification of X (so g : X -> A
is a map such that etc.) Say (h, B) is another
compactification of X. Then (g,A) "compactification-
embeds" in (h, B) if there exists an
embedding f : A -> B such that f(g(x)) = h(x)
for all x in X. (If we pretend that X s literally
a suubset of A and of B then the condition is that
f fix every point of X.)

Which come to think of it surely never happens
unless f is a homeomophism: f(A) is compact,
hence closed, and contains X, so f(A) is dense,
hence f(A) = B and standard blah blah shows
the inverse of f is continuous.

So. Given the _relevant_ notion of one compactification
embedding in another, it never happens unless the
two compactifications are equivalent. Embedding
is simply not what we're talking about when we talk

>
>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