Exercise 22, § 11

a) Show that in an extremally disconnected space every open subspace and every dense subspace is extremally disconnected.

b) Let $X$ be a Hausdorff space in which every point has a neighbourhood which is an extremally disconnected subspace of $X$. Show that $X$ is extremally disconnected.

c) Give an example of a Hausdorff space $X$ which is not extremally disconnected but which has an extremally disconnected dense open subspace (paste together two extremally disconnected spaces).

d) Let $X$ be a countable discrete space, $\tilde{X}$ the (compact) space of ultrafilters on $X$ (§ 9, Exercise 26). Let $(A_n)_{n \in \mathbf{N}}$ be a countably infinite partition of $X$ into infinite subsets; in the closed subspace $Y = \tilde{X} - X$ of $\tilde{X}$, the sets $B_n = \tilde{X}_n \cap Y$ are open and closed and pairwise disjoint. Let $B = \bigcup_n B_n$; show that the closure $\overline{B}$ of $B$ in $Y$ is not open in $Y$, and hence that $Y$ is not extremally disconnected (*). (Use reductio ad absurdum: using Exercise 20, let $C$ be an open and closed subset of $\tilde{X}$ such that $C \cap Y = \overline{B}$; consider a point $x_n$ in each of the sets $C \cap A_n$, let $\overline{J}$ be the closure of the set $J$ of points $x_n$, and show that $\overline{J}$ does not meet $B$).

(*) Exercises 21 h) and 22 d) are due to R. Ricabarra.