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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 StoneCech in relation to one point compactification.
He says the latter is the "minimal" compactification of a space whereas the StoneCech compactification is maximal in a sense described by Exercise 4 in section 53
"Let Y be an arbitrary comactification of X; let B(X) be the StoneCech 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 StoneCech 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.

