Exercise 21, § 11

a) Show that the following conditions on a topological space $X$ are equivalent: (i) for each open set $U$ in $X$, $\overline{U}$ is open; (ii) for each closed set $F$ in $X$, $\dot{F}$ is closed; (iii) for each disjoint pair of open sets $U, V$ in $X$, we have $\overline{U} \cap \overline{V} = \varnothing$. A Hausdorff space which satisfies these conditions is said to be extremally disconnected; it is then totally disconnected. The rational line is totally disconnected but not extremally disconnected.

b) A Hausdorff space is extremally disconnected if and only if the associated semi-regular space ($§ 8$, Exercise 20) is extremally disconnected.

c) Let $X$ be a Hausdorff space and let $A$ be a dense subset of $X$. Show that if $A$ is extremally disconnected and if the topology of $X$ is A-maximal ($§ 3$, Exercise 11), then $X$ is extremally disconnected.

d) Deduce from b) and c) that, if $X$ is any infinite set, the (compact) ultrafilter space of $X$ is extremally disconnected (cf. $§ 9$, Exercises 26 and 24) (**).

e) Let $X$ be a Hausdorff space with no isolated points. Show that the topology of $X$ is quasi-maximal ($§ 2$, Exercise 6) if and only if $X$ is

() This condition is equivalent to saying that $Y - X$ is totally disconnected (Chapter II, $§ 4$, no. 4, Proposition 6). (*) For examples of extremally disconnected compact spaces with no isolated points, see Chapter II, $§ 4$, Exercise 12 b).

submaximal (§ 8, Exercise 22) and extremally disconnected. (To show that the condition is sufficient, use Exercise 22 e) of § 8 and prove that every closed subset of $X$ which has no isolated points is open).

f) Show that every ultraregular space (§ 8, Exercise 25) is extremally disconnected, but that an extremally disconnected compact space with no isolated points is not ultraregular (cf. § 9, Exercise 27).

g) Show that an extremally disconnected semi-regular space (§ 8, Exercise 20) is regular (cf. Chapter II, § 4, Exercise 12).

h) Show that in an extremally disconnected space there is no convergent sequence which has an infinite number of distinct terms [argue by contradiction by constructing recursively two sequences $(U_n), (V_n)$ of mutually disjoint open sets such that if $U = \bigcup_n U_n$ and $V = \bigcup_n V_n$ then $U \cap V = \emptyset$ but $\overline{U} \cap \overline{V} \neq \emptyset$].

i) Show that in an extremally disconnected space $X$ a non-isolated point $a$ cannot have a fundamental system of simultaneously open and closed neighbourhoods which is well-ordered (with respect to the relation $\supset$). (Use reductio ad absurdum, as in h).