Exercise 7, § 3

Let B be a normal topological space. Let $(U_i)_{i\in I}$ be a locally finite open covering of B. For every pair $(i, j)$ of elements of I and every $b\in U_i\cap U_j$, let $V_{i,j}(b)$ be a neighbourhood of $b$ contained in $U_i\cap U_j$. Prove that there exists a family $(V(b))_{b\in B}$ satisfying the following properties:

(i) For every $b\in B$, the subset $V(b)$ is an open neighbourhood of $b$;

(ii) If $b\in U_i\cap U_j$, then $V(b)\subset V_{i,j}(b)$;

(iii) If $V(a)$ and $V(b)$ have a point in common, there exists $i\in I$ such that $V(a)$ and $V(b)$ are both contained in $U_i$.

(Introduce an open covering $(U'_i)_{i\in I}$ of B such that $\overline{U'_i}\subset U_i$ for every $i$. Construct a family $(V(b))_{b\in B}$ satisfying conditions (i) and (ii) and such that, if $V(b)$ meets $\overline{U'_i}$, then $b\in \overline{U'_i}$. Prove that property (iii) is then satisfied.)