
 StoneWeierstrass theorem 
  The StoneWeierstrass theorem generalizes the Weierstrass approximation theorem in two directions: instead of the compact interval [a,b], an arbitrary compact Hausdorff space K is considered, and instead of the algebra of polynomial functions, approximation with elements from more general subalgebras of C(K) is investigated. 
  Further, there is a generalization of the StoneWeierstrass theorem to noncompact Tychonoff spaces, namely, any continuous function on a Tychonoff space space is approximated uniformly on compact sets by algebras of the type appearing in the StoneWeierstrass theorem and described below. 
  As a consequence of the Weierstrass approximation theorem, one can show that the space C[a,b] is separable: the polynomial functions are dense, and each polynomial function can be uniformly approximated by one with rational coefficients; there are only countably many polynomials with rational coefficients. 
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