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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 14, 2013 11:53 PM

On 3/14/2013 9:49 PM, William Elliot wrote:
> Let (g,Y) be a Cech Stone compactification of S.
> If (f,X) is a compactification of S, does X embed in 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?
>
> 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?
>

Munkres characterizes Stone-Cech in relation to
one point compactification.

He says the latter is the "minimal" compactification
of a space whereas the Stone-Cech compactification
is maximal in a sense described by Exercise 4 in
section 5-3

"Let Y be an arbitrary comactification of X; let B(X)
be the Stone-Cech compactification. Show there is a
continuous surjective closed map g:B(X) -> Y that
equals the identity on X

[This exercise makes precise what we mean by saying
B(X) is the 'maximal' compactification of X. If you
are familiar with quotient spaces, you will recognize
that g is a quotient map. Thus every compactification
of X is equivalent to a quotient space of B(X).]"

The question you ask is precisely Munkres definition:

"A compactification of a space X is a compact Hausdorff
space Y containing X such that X is dense in Y."

"In order to have a compactification, X must be
a completely regular space"

The Stone-Cech compactification is based on a cube
such that each component of the cube is an interval

I_a = [glb(f_a(X)),lub(f_a(X))]

formed from a bounded continuous real valued function
on a completely regular space.

The cube is the product of all such intervals (all
such functions on the space).

Define h: X -> Pi_a I_a

where h(x)=(f_a(x))

By Tychonoff's theorem, the cube is compact.
Because X is completely regular, the collection of
functions separates points on X. This, makes h an
imbedding.

Munkres goes on to prove a few uniqueness conditions
that are true for the compactification derived from
this imbedding involving extensions of the original
bounded continuous functions on X to continuous
functions on B(X). Any two compactifications satisfying
these extension properties are equivalent up to
homeomorphim.

Hope that helps.

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