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Proof of Stone-Wierstrass theorem

  • Thread starter talolard
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Homework Statement



During the proof of the Stone-Wierstrass theorem we define a particular set and claim it is open. I can't figure out how to show it is open.
I wrote out the proof until that point to show how it is constructed, I apologize for the long windedness.

Assume that X is a compact hausdorf space[tex] A\subset C(X) [/tex]is an algebra that sperates points and has a continuosu function in it. then [tex]\overline{A}=C(X).[/tex]

Proof:

Since A seperates points then for any [tex]a,b\in X[/tex] there exists [tex] h\in A [/tex] such that [tex] h(a)\neq h(b) [/tex]. define [tex] g(x)=\frac{h(x)-h(a)}{h(b)-h(a)}. [/tex] Then [tex] g(a)=0 [/tex]and [tex] g(b)=1[/tex].

for any function [tex] f\in A [/tex] define [tex] f_{\left(a,b\right)}(x)=\left(f(b)-f(a)\right)g(x)+f(a)\text{then }f_{\left(a,b\right)}(a)=f(a)\text{ and }f_{\left(a,b\right)}(b)=f(b) [/tex].

Set [tex]\epsilon>0 [/tex] and define [tex] U_{\left(a,b\right)}=\left\{ x\quad|f_{\left(a,b\right)}(x)<f(x)+\epsilon\right\} .[/tex]

My question is how do i gaurantee that [tex] U_{\left(a,b\right)} [/tex] is open? My professor took it for granted. I'm sure it has to do with the pre-image of f but i can't write it out.

Homework Equations





The Attempt at a Solution


Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
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With such a questions, you have to rewrite [tex]U_{(a,b)}[/tex] in such a way so that it is clear that it's open. In this case:

[tex]U_{(a,b)}=\{x~\vert~f_{(a,b)}(x)<f(x)+\epsilon\}=\{x~\vert~(f_{(a,b)}-f)(x)<\epsilon\}=(f_{(a,b)}-f)^{-1}(]-\infty,\epsilon[)[/tex].
 

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