Editing Stone–Weierstrass theorem (section)

=== Applications ===
As a consequence of the Weierstrass approximation theorem, one can show that the space {{math|C[''a'', ''b'']}} is [[separable space|separable]]: the polynomial functions are dense, and each polynomial function can be uniformly approximated by one with [[rational number|rational]] coefficients; there are only [[countable|countably many]] polynomials with rational coefficients. Since {{math|C[''a'', ''b'']}} is [[metrizable space|metrizable]] and separable it follows that {{math|C[''a'', ''b'']}} has [[cardinality]] at most {{math|2<sup>ℵ<sub>0</sub></sup>}}.  (Remark: This cardinality result also follows from the fact that a continuous function on the reals is uniquely determined by its restriction to the rationals.)