Exercise 7, § 2

Let $X$ be a non-empty linearly ordered set, endowed with the topology $\mathfrak{T}_0(X)$.

a) If $X$ is connected, show that it has the following two properties:

α) Every non-empty subset of $X$ which is bounded above has a least upper bound (if $A$ is non-empty and bounded above, and if $B$ is the set of upper bounds of $A$, and $C$ the set of lower bounds of $B$, show that $B \cup C = X$ and that $B$ and $C$ are closed).

β) The set $X$ is without gaps, in other words (Set Theory, Chapter III, § 1, Exercise 18) every open interval $[a, b[\ (a < b)$ of $X$ is non-empty (use the argument of Theorem 4).

b) Show conversely that if $X$ has properties α) and β), then $X$ is connected. (Show first that every closed interval $[a, b]$ is connected: assuming that this interval is the disjoint union of two non-empty closed sets $A$ and $B$, and that $a \in A$, consider the greatest lower bound of $B$ and hence obtain a contradiction).

c) Show that if $X$ is connected it is locally compact and locally connected, and that the only connected subsets of $X$ are the intervals (bounded or unbounded).