Lemma 1

Each of properties (i) to (iv) of theorem 2 implies property (PCV).

Let $(U_i)_{i\in I}$ be an open covering of A by open sets of B. Under hypothesis (i), there exists a paracompact neighbourhood V of A contained in the union of the $U_i$, an open covering $(U'_j)_{j\in J}$ of the space V, locally finite and finer than the covering $(V\cap U_i)_{i\in I}$, and an open covering $(U''_j)_{j\in J}$ of the space V such that for every $j\in J$, the closure $F_j$ of $U''_j$ in V is contained in $U'_j$ (TG, IX, p. 49, prop. 4 and p. 48, cor. 1 to th. 3). Hence property (PCV) in this case.

Condition (ii) implies condition (i) by Corollary 2 of TG, IX, p. 50. Analogously, condition (iii) implies condition (i): in fact, if B is metrisable, every neighbourhood of A is metrisable, hence paracompact (TG, IX, p. 51, th. 4).

Lastly, suppose that condition (iv) is satisfied, and let us show that property (PCV) holds. Since A is compact, it is enough to consider the case of a finite covering $(U_i)_{0\leqslant i\leqslant n}$. We then construct by induction open sets $V_0, . . . ,V_n$ of B satisfying the following conditions for $0\leqslant i\leqslant n:$

Suppose that $V_0, . . . ,V_{r-1}$ have been constructed satisfying the above conditions for $0\leqslant i\leqslant r-1$, and let us construct $V_r$. The sets K = $A\cap \complement U_r$ and $L = A\cap \complement (V_0\cup \cdots \cup V_{r-1}\cup U_{r+1}\cup \cdots \cup U_n)$ are closed in A, hence compact. By virtue of $(\alpha )$, they are disjoint. By hypothesis, for every point $a$ of L and every point $b$ of K, there exist disjoint neighbourhoods of $a$ and $b$ in B. By Lemma 2 below, there exist disjoint open sets $V_r$ and W in B such that $L\subset V_r$ and $K\subset W$. From the inclusions $L\subset V_r$ and $A\subset V_0\cup \cdots \cup V_{r-1}\cup U_r\cup \cdots \cup U_n$ and from the definition of L, it follows that $(\alpha )$ holds for $i=r$. On the other hand, we have $\overline{V_r}\cap K =\emptyset$, hence $\overline{V_r}\cap A\subset U_r$.

The set $M =\bigcup_{0\leqslant i\leqslant n}(V_i\cap \complement U_i)$ is closed and does not meet A by $(\beta )$. By $(\alpha )$, the set $V =((\bigcup_{0\leqslant i\leqslant n}\overline{V_i})-$ M is a neighbourhood of A in B. For $i= 0, . . . , n$, put $F_i= V\cap \overline{V_i}:$ this is a closed subset of V, contained in $U_i$. The family $(F_i)_{0\leqslant i\leqslant n}$ is a covering of V, whence property (PCV).