What axiom of separation is needed here? Sorry, I didn't know how to provide a more concrete title.
Let $A_1,\cdots, A_{n+1}$ be a family of nonempty closed sets such that $\bigcap A_i = \emptyset$, and that $A_{n+1} \not \subseteq A_1, \cdots A_n$. What, if any, axioms of separation are sufficient to ensure that $A_{n+1}$ can always be broken down as a union of nonempty closed sets $B_1 \cup \cdots \cup B_n$, where $B_i \cap A_i = \emptyset $?
Edit: In normal spaces, $n=2$ follows quite easily from Urysohn's lemma.
 A: We shall follow Paul Sinclair’s reformulation. The case $n=2$ is equivalent to an axiom $T_{4-}$: “any two disjoint closed subsets have disjoint open neighborhoods”. This axiom is a bit weaker than normality and can be understood as $T_4$ without $T_1$.
Let us show that $T_{4-}$ is exactly the required separation axiom, by induction with respect to $n$. A base case $n=2$ was already considered. Suppose that the induction hypothesis holds for $n$ and let $C_1,\dots, C_{n+1}$  are closed subsets of a $T_{4-}$ space $X$ such that $\bigcap_{i=1}^{n+1} C_i=\varnothing$. For each natural $i$ from $1$ to $n$ put $C’_i=C_i\cap C_{n+1}$. Then $C’_1,\dots, C’_n$  are closed subsets of $X$ such that $\bigcap_{i=1}^{n} C’_i=\varnothing$. By the induction hypothesis there exist open sets $U’_i\supset C_i$ for each natural $i$ from $1$ to $n$ such that $\bigcap_{i=1}^{n} U’_i=\varnothing$. Since for each natural $i$ from $1$ to $n$, $C_i\cap C_{n+1}=C’_i\subset U’_i$, $C_i\setminus U’_i$ and $C_{n+1}$ are disjoint closed subsets of $X$. Therefore there exists disjoint open subsets $V_i\supset C_i\setminus U’_i$ and $W_i\supset C_{n+1}$ of $X$. Remark that $U_i=U’_i\cup V_i\supset C_i$ and $U_{n+1}=\bigcap_{i=1}^{n} W_i\supset C_{n+1}$. Finally, we have $$\bigcap_{i=1}^{n+1} U_i=\bigcap_{i=1}^{n} (U’_i\cup V_i)\cap W_i =\bigcap_{i=1}^{n} U’_i\cap W_i\subset \bigcap_{i=1}^{n} U’_i=\varnothing.$$
