Let $C$ be a set of all continuous functions that map from a compact set to the reals. Is the set of all polynomials dense in $C$, with respect to the sup metric?

Can I show that by using the Stone-Weierstrauss Theorem? Since I can find a sequence of polynomials that converge uniformly to any continuous function, if I take the closure of the set of polynomials, do I get the entire set of continuous functions?

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    $\begingroup$ No, because Stone-Weierstrass only applies to compact sets. $e^x$ is not uniformly approximable by polynomials. $\endgroup$ – user296602 Dec 7 '17 at 17:17
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    $\begingroup$ Dense with respect to which topology? $\endgroup$ – José Carlos Santos Dec 7 '17 at 17:18
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    $\begingroup$ @user296602 ... which is not a metric on the set of continuous functions, by the way. $\endgroup$ – user228113 Dec 7 '17 at 17:24
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    $\begingroup$ @G.Sassatelli Is that why I need compactness? $\endgroup$ – tryingtosolve Dec 7 '17 at 17:26
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    $\begingroup$ Updated the question again. $\endgroup$ – tryingtosolve Dec 7 '17 at 17:32

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