Exercise 17, § 2

Let $\beta \mathbf{N}$ be the Stone-Čech compactification of the discrete space $\mathbf{N}$ ($\S$ 1, Exercise 7).

a) Show that there are two countably compact subspaces $A, B$ of $\beta \mathbf{N}$ such that $A \cap B = \mathbf{N}$ and $A \cup B = \beta \mathbf{N}$. [Let $\aleph_\alpha = 2^{2^{\text{Card}(\mathbf{N})}}$; by virtue of $\S$ 1, Exercise 12 b), there is a bijection $\xi \to S_\xi$ of the ordinal $\omega_\alpha$ (Set Theory, Chapter III, $\S$ 6, Exercise 10) onto the set of countably infinite subsets of $X$. By transfinite induction, define two injective mappings $\xi \to x_\xi, \xi \to y_\xi$ of $\omega_\alpha$ into $\beta \mathbf{N} - \mathbf{N}$ such that $x_\xi \in \overline{S}_\xi, y_\xi \in \overline{S}_\xi$ for each $\xi$, and such that if $P$ (resp. $Q$) is the set of the $x_\xi$ (resp. $y_\xi$) we have $P \cap Q = \varnothing, P \cup Q = \beta \mathbf{N} - \mathbf{N}$; for this, make use of $\S$ 1, Exercise 12 b) and c). Then show that $A = P \cup \mathbf{N}$ and $B = Q \cup \mathbf{N}$ satisfy the conditions of the question.]

b) Show that the product space $A \times B$ is not countably compact (note that the intersection of $A \times B$ with the diagonal of $\beta \mathbf{N} \times \beta \mathbf{N}$ is an infinite discrete closed subspace of $A \times B$).