Exercise 9, § 4

a) Let $X$ be a paracompact space in which every point has a countable fundamental system of neighbourhoods, and let $Y$ be a countably compact normal space ($§ 2$, Exercise 14). Show that $X \times Y$ is normal. [Let $A, B$ be two disjoint closed subsets of $X \times Y$; for each $x \in X$, show that there is a neighbourhood $U_x$ of $x$ in $X$ and two open sets $V_x, W_x$ in $Y$ such that $\overline{V_x} \cap \overline{W_x} = \emptyset$ and such that for each $z \in U_x$ we have $A(z) \subset V_x$ and $B(z) \subset W_x$; to prove this, argue by contradiction. Let $(T_\lambda)_{\lambda \in L}$ be a locally finite open covering of $X$, finer than the covering $(U_x)_{x \in X}$, and let $(f_\lambda)$ be a partition of unity subordinate to $(T_\lambda)$; for each $\lambda \in L$, let $x_\lambda$ be such that $T_\lambda \subset U_{x_\lambda}$ and let $g_\lambda$ be a continuous mapping of $Y$ into $[0, 1]$ which is equal to 0 on $\overline{V}_{x_\lambda}$ and to 1 on $\overline{W}_{x_\lambda}$; consider the function $\sum f_\lambda(x) \ g_\lambda(y)$ on $X \times Y$.]

b) Let $Y = [a, b[$ be the locally compact space defined in Chapter I, § 9, Exercise 12, and let $X = [a, b]$ be the compact space obtained by adjoining a point at infinity to $Y$. $Y$ is normal (Exercise 4) and countably compact (§ 2, Exercise 15), but the product $X \times Y$ is not normal (use Chapter II, § 4, Exercise 4).