Exercise 7, § 4

Let $X$ be a linearly ordered set.

a) Let $X_1$ be the completion of $X$ (Set Theory, Chapter III, § 1, Exercise 15). $X_1$ is a linearly ordered set whose elements are the subsets $A$ of $X$ such that; (i) if $x \in A$ and $y \leq x$, then $y \in A$; (ii) if $A$ has a least upper bound in $X$, this bound belongs to $A$ [which is equivalent to saying that $A$ is closed in $\mathfrak{T}_0(X)$]; the elements of $X_1$ are ordered by inclusion. $X_1$ is compact with respect to the topology $\mathfrak{T}_0(X_1)$ [§ 2, Exercise 6 a)]. The mapping $x \to ]\leftarrow, x]$ of $X$ into $X_1$ is strictly order-preserving, and if we identify $X$ with its image under this mapping, then $X$ is dense in $X_1$ and $\mathfrak{T}_0(X)$ is induced by $\mathfrak{T}_0(X_1)$. $X_1$ is connected if and only if $X$ is without gaps (§ 2, Exercise 7).

b) For each cardinal $c$, give an example of a linearly ordered set $X$ which is connected in the topology $\mathfrak{T}_0(X)$ and is such that every fundamental system of neighbourhoods of any point of $X$ has cardinal $\geq c$ [use a) and the method of § 1, Exercise 4 b)].

c) With the notation of a), let $X_2$ be the subset of the lexicographic product $X_1 \times \{-1, 0, 1\}$ which is the complement of the set consisting of: (i) the points $(x, 0)$ where $x \notin X$; (ii) the points $(x, 1)$ where $x \in X$ and the set of all $y > x$ in $X$ has a smallest element; (iii) the points $(x, -1)$ where $x \in X$ and the set of all $y < x$ in $X$ has a greatest element. Show that $X_2$, endowed with the topology $\mathfrak{T}_0(X_2)$, is compact and totally disconnected, that $X$ can be identified, by means of the strictly order-preserving mapping $x \to (x, 0)$, with a dense subset of $X_2$, and that the topology induced by $\mathfrak{T}_0(X_2)$ on $X$ is discrete.

d) Let $X'$ be a linearly ordered set containing $X$ which induces the given order structure on $X$, is compact in the topology $\mathfrak{T}_0(X')$ and is such that $X$ is dense in $X'$ with respect to this topology. Show that there is a continuous order-preserving surjective mapping $f : X' \to X_1$ and a continuous order-preserving surjective mapping $g : X_2 \to X'$, both of which induce the identity mapping on $X$.