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...

In this paper, localic upper, respectively lower continuous chains over a locale are defined. A localic Katětov-Tong insertion theorem is given and proved in terms of a localic upper and lower continuous chain. Finally, the localic Urysohn lemma and the localic Tietze extension theorem are shown as applications of the localic insertion theorem.

Let $L_c\left(X\right)=\{f\in C\left(X\right):\overline{C_f}=X\}$, where $C_f$ is the union of all open subsets $U\subseteq X$ such that $\left|f\left(U\right)\right|\le {\aleph }_0$. In this paper, we present a pointfree topology version of $L_c\left(X\right)$, named ${ℛ}_{\ell c}\left(L\right)$. We observe that ${ℛ}_{\ell c}\left(L\right)$ enjoys most of the important properties shared by $ℛ\left(L\right)$ and ${ℛ}_c\left(L\right)$, where ${ℛ}_c\left(L\right)$ is the pointfree version of all continuous functions of $C\left(X\right)$ with countable image. The interrelation between $ℛ\left(L\right)$, ${ℛ}_{\ell c}\left(L\right)$, and ${ℛ}_c\left(L\right)$ is examined. We show that $L_c\left(X\right)\cong {ℛ}_{\ell c}\left(𝔒\left(X\right)\right)$ for any space $X$. Frames $L$ for which ${ℛ}_{\ell c}\left(L\right)=ℛ\left(L\right)$ are characterized.