FOM: New member/ Stone-Weierstrass theorem

Andrej Bauer Andrej.Bauer at
Thu May 10 20:25:34 EDT 2001

> What prompted me to subscribe now is a question by one of my fellow
> mathematicians here in Dublin. This guy is writing a book on analysis and at
> some point he introduces the Stone-Weierstrass theorem. The proof he gives
> in his book uses very heavily the Axiom of Choice, but he read somewhere that
> it is possible to give a proof avoiding the Axiom. I wonder if somebody on
> this list might know about this or would be able to suply some source for
> this.
> For completeness i insert a statement of the Stone-Weierstrass theorem (i hope
> i have it correct): Let X be a compact Hausdorff space and let A be an algebra
> of complex continous functions on H which is:
> a) closed in the ring C of all complex continous functions
> b) contains all the constant functions
> c) is closed under complex conjugation 
> Then A=C.

Peter Johnstone, "Stone Spaces", Cambridge University Press, Chapter
IV, Paragraph 5.4, page 154, he proves the Stone-Weierstrass Theorem
for real-valued continuous functions on a compact Hausdorff space (but
as he says on page 153, the same proof works for the complex-valued
continuous functions).

In the book, all theorems that use the Axiom of Choice are marked by a
star (*), and I do not see one next to the Stone-Weierstrass Theorem,
and neither does the proof seem to use it.

Andrej Bauer
Mittag-Leffler Institute

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