Exercise 15, § 2

a) Let $X$ be a connected Hausdorff space such that, for each $x \in X$, $C\{x\}$ has exactly two components. Show that, if $K$ is either of these components, $K = K \cup \{x\}$ [Chapter I, § 11, Exercise 4 a)]. Let $x, y$ be two distinct points of $X$, let $A$ and $B$ be the components of $C\{x\}$ and let $A', B'$ be the components of $C\{y\}$. Show that one of the two sets $A, B$ is contained in $A'$ or in $B'$.

b) Let $x_0$ be a point of $X$ and let $A(x_0), B(x_0)$ be the components of $C\{x_0\}$. For each $x \neq x_0$ there is exactly one of the two components of $C\{x\}$ which is either contained in $A(x_0)$ or contains $A(x_0)$; denote this component by $A(x)$. Show that the relation $A(x) \subset A(y)$ is a linear order relation compatible (Exercise 14) with the topology of $X$.

c) Extend the conclusion of b) to the case where $C\{x\}$ has two components for all $x \in X$ except for one point $a \in X$, for which $C\{a\}$ is connected.

*d) Let $X$ be the subspace of $\mathbf{R}^2$ consisting of the points $a = (0, 1)$, $b = (0, -1)$ and all pairs $(x, y)$ such that $x \neq 0$ and $y = \sin(1/x)$. Show that $X$ is connected in the topology $\mathfrak{T}$ induced on $X$ by the topology of $\mathbf{R}^2$, that $C\{a\}$ and $C\{b\}$ are connected and that, for each point $x$ of $X$ other than $a$ and $b$, $C\{x\}$ has exactly two components; but that there exists no linear ordering on $X$ compatible with the topology $\mathfrak{T}$ *.