FOM: New member/ Stone-Weierstrass theorem

[Karim Zahidi]

Hello

My name is Karim Zahidi and currently I hold a one-year position as Postdoc
fellow at the Dept. of Mathematics at University College Dublin (Ireland).  I
did my ph d research at Ghent University (Belgium) on the topic of
undecidability of various existential theories of rings and fields which come
up in number theory (i.e. extending the negative solution of Hilbert's Tenth
Problem by Davis, Matijaevich Putnam and Robinson to these fields and rings).


 
I just subscribed to the fom list, after browsing through the archives for
several months. While I find many of the discussions I encountered interesting
I always refrained from subscribing (basically the reason is time and the fact
that I'm not really a specialist in fom ).

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.

Looking forward to interesting discussions on this list.

Karim Zahidi