Q: O. Frink's characterization of completely regular spaces Def: Suppose $X$ is topological space and $\mathcal{B}$ is a base for it. We say, that $\mathcal{B}$ is normal base, if following properties hold:
a. For any $x \in X$ and $A \in \mathcal{B}$, with $x \in A$, there exist $A' \in \mathcal{B}$, such that $x \notin A'$ and $A \cup A' = X$.
b. If $U$ and $V$ are open sets from $\mathcal{B}$, such that $U \cup V = X$, than there exist $U'$ and $V'$, disjoint sets from $\mathcal{B}$, such that $X\setminus U \subseteq V'$ and $X\setminus V \subseteq U'$.
It is well known that the space is completely regular iff it has normal base. This characterizations first published in Orrin Frink's paper "compactifications and semi-normal spaces".
It's easy part to prove that every completely regular space has normal base, because the set of co-zero sets is such a base, which is well-known to be a base for a completely regular space.
Inverse part, that every space which has normal base is completely regular, maybe is also easy, because there is a hint that this is actually the rewriting of proof of Urysohn lemma. But, however I still have a problem with this proof and can't construct the scheme like from Urysohn's lemma. Please, help me if you can.
