Exercise 26, § 4

a) Show that every pseudo-compact normal space $X$ (§ 1, Exercise 22) is countably compact (§ 2, Exercise 14). [Show that if $(x_n)$ is a sequence of distinct points of $X$ with no cluster point, then there is a continuous finite real-valued function $f$ on $X$ such that $f(x_n) = n$ for all $n$.]

b) Let $Y = [a, b[$ be the locally compact space defined in Chapter I, § 9, Exercise 12, and let $Y_0$ be the compact space obtained by adjoining to $Y$ a point at infinity (which may be identified with $b$). Let $c$ be the smallest element of $Y$ such that $[a, c[$ is infinite, and let $Z$ be the compact subspace $[a, c]$ of $Y$. Let $X$ be the complement of the point $(b, c)$ in the compact product space $Y_0 \times Z$. Show that $X$ is locally compact but not normal [cf. Exercise 12 a)]; that $X$ is pseudo-compact but not countably compact. Give an example of a closed subspace of $X$ which is not pseudo-compact. Also, give an example of a lower semi-continuous function on $X$ which does not attain its greatest lower bound [cf. § 2, Exercise 14 i)].