Corollary

Suppose that $H$ is such that, whenever $u \in H$ and $v \in H$, we have $\sup(u, v) \in H$ and $\inf(u, v) \in H$; and is such that, given any two distinct points $x, y$ of $X$ and any two real numbers $\alpha, \beta$, there is a function $g \in H$ such that $g(x) = \alpha$ and $g(y) = \beta$. Then every continuous real-valued function on $X$ can be uniformly approximated by functions belonging to $H$.