Exercise 8, § 8

a) Let $X$ be a Hausdorff (resp. regular) space which is the sum of a family $(X_i)$ of subspaces, and let $X/R$ be the quotient space obtained by pasting together the $X_i$ along closed subsets $A_{i,x}$ by means of homeomorphisms $h_{x,i}$ ($§ 3$, no. 4); suppose also that for each index $i$ there is only a finite number of indices $x$ such that $A_{i,x} \neq \varnothing$. Under these conditions, show that $X/R$ is Hausdorff (resp. regular).

\* b) Show that the space $X/S$ of Exercise 7 can be obtained by pasting together two regular subspaces along two open sets. \*