Exercise 13, § 4

. Prove that the product $X \times Y$ is normal. (Let A, B be two closed parts of $X \times Y$ having no point in common; for every $x \in X$, prove that there exists a neighbourhood $U_x$ of x in X and two open sets $V_x, W_x$ in F, such that $\overline{V_x} \cap \overline{W_x} = \varnothing$ and that for every $z \in U_x$, one has $A(z) \subset V_x$ and $B(z) \subset W_x$; to see this one reasons by contradiction. Let $(T_\lambda)_{\lambda \in L}$ be a locally finite open covering of X, finer than $(U_x)_{x \in X}$, and let $(f_\lambda)$ be a partition of unity subordinate to $(T_\lambda)$; for every $\lambda \in L$, let $x(\lambda)$ be such that $T_\lambda \subset U_{x(\lambda)}$, and let $g_\lambda$ be a continuous mapping from Y into $[0, 1]$, equal to 0 in $\overline{V}_{x(\lambda)}$, to 1 in $\overline{W}_{x(\lambda)}$; consider in $X \times Y$ the function $\sum_\lambda f_\lambda(x) g_\lambda(y)$). b) Let $X = [a, b[$ be the locally compact space defined in exerc. 12 of I, p. 106, $Y = [a, b]$ the compact space obtained by adjunction to X of a point at infinity; X is normal (IX, p. 102, exerc. 8) and semi-compact (IX, p. 94, exerc.