Exercise 8, § 4

Let $(X_i)_{1 \leq i \leq n}$ be a finite family of compact spaces, let $X = \prod_{i=1}^n X_i$ be their product, and for each index $i$ let $F_i$ be a closed subset of $X_i$. Show that every continuous real-valued function on $X$ which vanishes on the union of the sets $F_i \times \prod_{j \neq i} X_j$ ($1 \leq i \leq n$) can be uniformly approximated by finite sums of functions of the form

where $u_i$ is a continuous real-valued function on $X_i$ which vanishes on $F_i$ ($1 \leq i \leq n$).