From: Daniel Giaimo <dgiaimo@rgiaimo.net>
Subject: Re: Did any body study C(X)?
Date: Tue, 26 Dec 2000 21:12:04 -0800
Newsgroups: sci.math.research
Summary: [missing]

"David Cohen_Steiner" <dcohen@sophia.inria.fr> wrote in message
news:91b1on$p4c$1@news-sop.inria.fr...
> A theorem of Gelfand and Kolmogorov says that if C(X) and C(Y) are
> isomorphic, then X and Y are homeomorphic, where X
> and Y are compact Hausdorff. (Dugundji, Topology, Ally&Bacon,1966 pp 289).

    In fact, not only are X and Y homeomorphic, but, more specifically,
for any compact Hausdorff space X, the mapping \mu:X -> MaxSpec(C(X))
sending a point x to the ideal of functions vanishing at that point is a
homeomorphism of X onto the maximal spectrum of C(X).  You can find a
nice proof of this in Exercise 26 of Chapter 1 of Atiyah & Macdonald's
"Introduction to Commutative Algebra".

--Daniel Giaimo