Exercise 10, § 1

a) Let $\mathfrak{M}$ be a finite set and let $R \{ x, y \}$ be a symmetric relation between elements $x, y$ of $\mathfrak{M}$. Suppose that there exist two distinct elements $a, b$ in $\mathfrak{M}$ with the following properties: there exist $x \neq a$ and $y \neq b$ such that $x$ (resp. $y$) is the only element $z \in \mathfrak{M}$ such that $R \{ a, z \}$ (resp. $R \{ b, z \}$) is true; and for each $t \in \mathfrak{M}$ other than $a$ and $b$, the set of all $z \in \mathfrak{M}$ such that $R \{ t, z \}$ is true is a set consisting of two elements, distinct from $t$. Show that under these conditions there exists a bijection $i \to x_i$ of an interval $[0, n]$ of $\mathbf{N}$ onto $\mathfrak{M}$ such that $x_0 = a, x_n = b$ and such that $R \{ x_{i-1}, x_i \}$ is true for $1 \leq i \leq n$ (define $x_i$ by induction on $i$).

b) In $\mathbf{R}^n$ ($n \geq 2$), identified with $\mathbf{R}^{n-1} \times \mathbf{R}$, let $L$ and $L'$ be two broken lines, defined respectively by two sequences $(x_i)_{0 \leq i \leq p}$ and $(x'_j)_{0 \leq j \leq q}$, with the following properties: (i) if we put $x_i = (y_i, z_i)$, $x'_j = (y'_j, z'_j)$, we have $z_0 = z'_0 = 0, z_p = z'_q = 1, 0 < z_i < 1$ for $1 \leq i \leq p - 1$ and $0 < z'_j < 1$ for $1 \leq j \leq q - 1$; (ii) the $p + q - 2$ numbers $z_i, z'_j$ ($1 \leq i \leq p - 1, 1 \leq j \leq q - 1$) are all distinct; (iii) two distinct sides of $L$ (resp. $L'$) have at most one point in common [which may or may not be one of the $x_i$ (resp. $x'_j$)]. Show that there exist two surjective piecewise linear mappings (Exercise 6) $u : I \to L, u' : I \to L'$, where $I$ is the interval $[0, 1]$ of $\mathbf{R}$, such that, if we put $u(t) = (v(t), \zeta(t))$, $u'(t) = (v'(t), \zeta'(t))$ ($t \in I$), we have $\zeta(t) = \zeta'(t)$ for all $t \in I$,

[Let $(a_k)_{1 \leq k \leq p+q-2}$ be the strictly increasing sequence formed by the $z_i$ and the $z'_j$ other than 0 and 1, and put $a_0 = 0, a_{p+q-1} = 1$; let $B_i$ denote the set of all $x = (y, z) \in \mathbf{R}^n$ such that

let $\mathfrak{M}$ denote the set of all pairs $\gamma = (C, C')$, where $C$ (resp. $C'$) is the intersection of the same $B_i$ with a side of $L$ (resp. $L'$), such that neither $C$ nor $C'$ consists of a single point; and let $R \{ \alpha, \beta \}$ denote the following relation between elements $\alpha, \beta$ of $\mathfrak{M}$: " $\alpha \neq \beta$; if $\alpha = (C_1, C'_1), \beta = (C_2, C'_2)$, then $C_1 \cap C_2$ and $C'_1 \cap C'_2$ are not empty, one of them consists of a single point, and if this point is not one of the $x_i$ (resp. $x'_j$) then $C_1$ and $C_2$ (resp. $C'_1$ and $C'_2$) are both contained in the same side of $L$ (resp. $L'$)". Show that a) is applicable to the relation $R.$]

c) In $\mathbf{R}^n$, identified with $\mathbf{R}^{n-1} \times \mathbf{R}$, let $K$ be a connected compact set such that $K \cap (\mathbf{R}^{n-1} \times \{0\})$ contains at least two distinct points. Show that if $K' = K - K$ (the set of all $x - y$, where $x \in K$ and $y \in K$), then $K' \cap (\mathbf{R}^{n-1} \times \{0\})$ contains a connected set which does not consist of a single point. [Use b), applying Chapter II, , Proposition 6 and Exercise 15.]