Exercise 14, § 2

Let $X$ be a connected Hausdorff topological space not consisting of a single point, and let $\Delta$ denote the diagonal of $X \times X$.

a) Show that there is no partition of $C\Delta$ into two open sets $D_1, D_2$ such that $D_1^{-1} = D_1$ and $D_2^{-1} = D_2$. [Begin by remarking that $D_1(x)$ and $D_2(x)$ are connected for each $x \in X$ (use Exercise 4 a) of Chapter I, § 11); deduce that, if $y \in D_1(x)$, we have $D_2(x) \times \{y\} \subset D_1$, and show that this implies that, for each $x \in X$, one or the other of the sets $D_1(x)$, $D_2(x)$ is empty; hence show that one or the other of the sets $D_1, D_2$ is empty, which is contrary to hypothesis.] Deduce that either $C\Delta$ is connected or else $C\Delta$ has exactly two components $A, B$ such that $B = A^{-1}$.

b) A linear ordering on $X$ is said to be compatible with the topology $\mathfrak{T}$ of $X$ if $\mathfrak{T}_0(X)$ is coarser than $\mathfrak{T}$. Deduce from a) that there exists such a linear ordering on $X$ if and only if $C\Delta$ is not connected, and that there are then exactly two such linear orderings. [With the notation of a), show that the order relation must be either $(x, y) \in A$ or $(x, y) \in B = A^{-1}$.] Show that the intervals, for these orderings, are connected in the topology $\mathfrak{T}$, and that they are the only connected subsets of $X$ with respect to $\mathfrak{T}$.

c) Show that if $X$ is either locally connected or locally compact in the topology $\mathfrak{T}$, and if there exists a linear ordering on $X$ compatible with $\mathfrak{T}$, then we must have $\mathfrak{T} = \mathfrak{T}_0(X)$. [If $X$ is locally connected, use b). If $X$ is locally compact, show that every compact neighbourhood of a point $x \in X$ is also a neighbourhood of $x$ with respect to $\mathfrak{T}_0(X)$, by considering the frontier of the neighbourhood].

d) Let $X$ be the subspace of $\mathbf{R}^2$ consisting of $(0, 0)$ and all pairs $(x, y)$ such that $x > 0$ and $y = \sin(1/x)$. Show that there exists a linear ordering on $X$ which is compatible with the topology induced on $X$ by the topology of $\mathbf{R}^2$, but that this topology is strictly finer than $\mathfrak{T}_0(X)$. \ Give an example of a connected Hausdorff space $X$, no point of which has a fundamental system of connected neighbourhoods, and on which there is a linear ordering compatible with the topology of $X$ [cf. Chapter I, § 11, Exercise 2 b)].