The cardinal functions of pseudocharacter, closed pseudocharacter, and character are used to examine H-closed spaces and to contrast the differences between H-closed and minimal Hausdorff spaces. An H-closed space $X$ is produced with the properties that $\left|X\right|>2^{2^{\psi \left(X\right)}}$ and $\overline{\psi }\left(X\right)>2^{\psi \left(X\right)}$.

For a Tychonoff space $X$, $C\left(X\right)$ is the lattice-ordered group ($l$-group) of real-valued continuous functions on $X$, and $C^{*}\left(X\right)$ is the sub-$l$-group of bounded functions. A property that $X$ might have is (AP) whenever $G$ is a divisible sub-$l$-group of $C^{*}\left(X\right)$, containing the constant function 1, and separating points from closed sets in $X$, then any function in $C\left(X\right)$ can be approximated uniformly over $X$ by functions which are locally in $G$. The vector lattice version of the Stone-Weierstrass Theorem is more-or-less equivalent...