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Let $\mathscr{A}$ be an algebra of bounded complex functions. (Or if necessary, continuous and domain of functions is compact)

Definition:

$\mathscr{B}$ is uniformly closed iff $f\in\mathscr{B}$ whenever $f_n\in \mathscr{B} (n=1,2,\cdot)$ and $f_n\rightarrow f$ uniformly.

$\mathscr{B}$ is the uniform closure of $\mathscr{A}$ iff $\mathscr{B}$ is the set of all functions which are limits of uniformly convergent sequences of members of $\mathscr{A}$.

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Let $\mathscr{B}$ be a uniform closure of $\mathscr{A}$.

How do i prove that $\mathscr{B}$ is uniformly closed in ZF?

Does Stone-Weierstrass theorem require choice since it is critical to prove Stone-Weierstrass Theorem?

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    How do you define the uniform closure?2012-12-17
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    $\mathscr{B}$ is the uniform closure of $\mathscr{A}$ if $\mathscr{B}$ is the set of all functions which are limits of uniformly convergent sequences of members of $\mathscr{A}$.2012-12-17
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    I just edited my question. Is there a term to distinguish these two definitions (one is on the post and the other is $\overline{\mathscr{A}}$)? The relation looks very similar to that between sequential continuity and $\epsilon-\delta$ continuity.2012-12-17

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