Exercise 6, § 2

a) Let $X$ be a non-empty linearly ordered set, endowed with the topology $\mathfrak{T}_0(X)$ (Chapter I, § 2, Exercise 5). Show that $X$ is compact if and only if each subset of $X$ has a least upper bound, in other words if and only if $X$ is a complete lattice (Set Theory, Chapter III, § 1, Exercise 11). (To show that the condition is necessary, argue as for Theorem 3; for sufficiency, consider a filter $\mathfrak{F}$ on $X$ and show that, if $A$ is the set of greatest lower bounds of sets of $\mathfrak{F}$, then the least upper bound of $A$ is a cluster point of $\mathfrak{F}$).

b) Give an example of a complete lattice $X$ which is not compact in the topology $\mathfrak{T}_0(X)$.