Exercise 11, § 4

a) Let $\Gamma$ be any set of subsets of a set $A$. Let $\Psi$ the set of subsets of $A$ of the form

where the $X_i$ are sets in $\Gamma$, $m$ is any integer $\geq 1$, and $p$ is any integer $\geq 0$. Show that the smallest clan $\Phi$ containing $\Gamma$ is the set of finite unions of sets in $\Psi$.

b) Let $\Delta$ be the set of finite intersections of sets in $\Gamma$. Show that, for every vector space $F$, the set of linear combinations (with coefficients in $F$) of characteristic functions of sets of the clan generated by $\Gamma$ is identical to the set of linear combinations of characteristic functions of sets in $\Delta$.

c) Let $X$ be a topological space, $\Gamma$ the set of compact subsets of $X$. Show that the clan generated by $\Gamma$ is identical to the set of finite unions of subsets of $X$ of the form $A \cap \mathbf{C} B$, where $A$ and $B$ are compact sets.