Lef $f$ be a defined and continuous function in $[a,b]$. Given $\epsilon>0,$ there exists a polynomial $P$ such that $\vert f(x)-P(x)\vert<\epsilon,$ for all $x\in[a,b].$
Let $X$ be a topological compact space. If $F$ is an algebra of $C(X)$ that separate points and contains the constant functions then $F$ is dense with respect to the uniform convergence in $C(X).$
How is that an algebra $F$ of $C(X)$ that separate points and contains the constant functions is the generalization of $P(x)$ ?
Can someone shed some light on this?