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I wonder if there is a principal reason to call this compactification Stone--Čech and not Čech--Stone. If a mathematical phenomenon is discovered by two people at the same time, the custom is to use their names in the alphabetical order. For some reason, the authors of this compactification are often referred in the opposite order. Any chance this is just bias of the English-speaking world? I would guess that the English-speaking mathematicians probably didn't hear about Čech's result until much later, when they were already accustomed to attributing this to Stone. If this is the case, then I believe that Wikipedia should stand as an example and present the names in the correct order. — Preceding unsigned comment added by 2A00:23C7:85B9:401:948D:585E:D448:EAAD ( talk) 10:02, 9 November 2020 (UTC)
I used this compactification just the other day and was able to reduce the volume of all things I won by 1.5 to 3 times. Thanks Wikipedia! —Preceding unsigned comment added by 70.162.83.30 ( talk) 02:18, 17 September 2007 (UTC)
The application to functional analysis strikes me as the most amazing piece of overkill and actually achieves very little. You can characterise the dual space of in a trivial manner - it's just the space of finitely additive finite measures on the underlying measure algebra. This is about a five line proof. Bringing the Stone Cech compactification and the Riesz representation theorem into it just complicates the issue for no apparent gain. David MacIver 18:00 21st April
> What if we change it for "the computation of the dual space of C_b(X)"; C_b(X) being the space of continous and bounded scalar-valued functions over a completely regular topological space? Do you think that would be more interesting or is it also "trivial"?
It is true that you can caracterize the dual space of in more elemental terms (though I wouldn't say "trivially") but I find it amazing that you can get a countably additive measure instead of a finitely additive one by enlarging its support. Anyway, I always find myself amazed by things other (smarter) mathematicians consider trivial.
Could you indulge me and write down that five line proof for me? J L 23:14, 22 April 2006 (UTC).
Sure. (Sorry for the delay. I don't have net access at the moment).
Define T : l^inf* -> { finitely additive measures on N } by Tf (A) = f(I_A) (where I_A is the indicator function of A). It's obvious that this is linear and gives a finitely additive measure. We just need to show that it's a bijection.
S, the set of simple functions (linear combinations of indicator functions) is dense in l^inf. If Tf = Tg then f and g agree on simple functions, and so by continuity they agree everywhere. Thus T is injective.
Now let m be a finitely additive measure. Without loss of generality we can assume it's positive. Then we can define f' on S by f'(I_A) = m(A) and extending linearly. Then (by positivity) we have ||f'|| <= m(N). So f' is continuous and can thus be extended to a linear functional on l^inf, say f. Obviously Tf = m.
QED
Ok, a bit more than five lines. Still short though.
David MacIver 10:15 9th May
The page says "every noncompact metric space contains a copy of betaN", which confuses me: (0,1) is a noncompact metric space of cardinality 2^omega - how can it contain a "copy" of betaN, which is of cardinality 2^2^omega?
I'm removing the sentence about "copy of Bw in any noncompact metric space", because it's just plain wrong. (Another example: the discrete omega is a noncompact metric space, with the discrete metric. How could it contain a copy of betaomega?) Aug 25 Jan Stary
I have a question about the proof given at the "Overkill" section.
Concerning the statement:
"Now let m be a finitely additive measure. Without loss of generality we can assume it's positive."
Why is it that we can assume that the measure `m' is positive? Is there some sort of Hahn descomposition for finitely additive measures?
I will be grateful if someone could clarify on this point or correct the proof. —Preceding unsigned comment added by 24.232.44.135 ( talk)
A section is named: An application: the dual space of the space of bounded sequences of reals. But we are speaking about the dual space of , i.e., about integers, not reals, aren't we? --
Kompik (
talk) 15:21, 31 January 2008 (UTC)
The construction using the Stone space of the complete Boolean algebra of all subsets of does not even mention the topology of , so it's probably wrong. I think, one has to use a different basis, namely the one consisting only of those sets with open . Does anyone know whether this is correct? —Preceding unsigned comment added by 129.187.111.36 ( talk) 11:15, 26 May 2009 (UTC)
Obviously compactification is of inherent interest, but are there any wellknown theorems in e.g. analysis, differential equations, etc, that hinges on the existence of a compactification? YohanN7 ( talk) 21:46, 7 July 2010 (UTC)
Good question -- over to you guys that know - chadnash
Where does the beta-notation come from?
Thanks!! 131.130.16.86 ( talk) 11:47, 24 January 2011 (UTC)
In this revision [1] the information that The Stone–Čech compactification was first considered by Tychonoff and later by Stone and Čech. The same information is given at given at Russian wikipedia ru:Компактификация Стоуна — Чеха.
I was trying to find whether this is true. According the sources I was able to find seems that it is not true.
Quote from Allen Shields: Years ago. The Mathematical Intelligencer, Volume 9, Number 2, 61-63, [2]
Also the nice diagram on p.27 of Russell C. Walker: The Stone-Čech compactification might be of interest in this connection. In fact, the only source supporting this claim, that I was able to find, is C. Wayne Patty: Foundations of topology By , p. 236:
Does anyone know about other sources supporting Tychonoff's precedence? If the claim is false, does anybody know where the misconception came from? -- Kompik ( talk) 22:43, 17 November 2011 (UTC)
Is there a reason, eluding me, for calling the unique map βf : βX → K a 'lift' of f : X → K ? Given the shape of the diagram, I would have rather said that βf extends f along the map X → βX. In the closing paragraph of that first section, it is properly termed the extension property of the functor β. 199.84.42.113 ( talk) 20:31, 29 February 2012 (UTC)