\[
  \mathcal{L}_r = \bigcup_{I \in S} (\widetilde{I} \smallsetminus \partial I), \qquad \mathcal{L}_s = \mathcal{L} \smallsetminus \mathcal{L}_r
\]

\[
  \widetilde{I} \cap \mathcal{L}_s \subset \partial I
\]

\[
    \widetilde{I}_{x,z},\ \widetilde{I}_{y,z} \subset \widetilde{I}, \qquad \partial I_{x,z} = \{x,z\}, \quad \partial I_{y,z} = \{y,z\}
  \]

\[
    \widetilde{I}_{x,z} \cap \widetilde{I}_{y,z} = \{z\}
  \]

\[
    \widetilde{I}' \cap \widetilde{I}'' = \struck{\ill{}}, \qquad \partial I' \cap \partial I'' = \{z\},
  \]

\[
  \begin{array}{lll}
    S \to \mathfrak{P}(\mathcal{L}), & I \mapsto \widetilde{I} & (\widetilde{I}\text{, ens. des lieux \emph{sur} le segment } I) \\
    S \to \mathfrak{P}_2(\mathcal{L}) & I \mapsto \partial I & (\partial I\text{, ens. des deux lieux \emph{extrémités} de } I)
  \end{array}
\]

\[
    \widetilde{I}_{x,z} \cap \widetilde{I}_{y,z} = \{z\}.
  \]

\[
    \widetilde{J} \subset \widetilde{I}_{x,z} \quad\text{ou}\quad \widetilde{J} \subset \widetilde{I}_{y,z}.
  \]

\[
  S_I \to \mathfrak{P}_2(\widetilde{I}), \qquad J \mapsto \partial J.
\]

\[
    \left\lbrace
    \begin{array}{l}
      I_{x,u} = J_{x,v} \quad (\text{donc } u = v \text{ et } u = v \in \widetilde{I}^{\circ} \cap \widetilde{J}^{\circ}) \\
      \underline{\underline{\text{ou}}} \quad \widetilde{I}_{x,u} \cap \widetilde{J}_{x,v} = \{x\}
    \end{array}
    \right.
  \]

\[
  B_x = S_x / R_x
\]

\[
    \widetilde{I}' \cap \widetilde{I}'' = \partial I' \cap \partial I'' = \{z\}, \quad \text{avec } z \text{ \emph{régulier}.}
  \]

\[
  \partial J \cap \partial I = \{y\} \qquad (\text{et } x \in \widetilde{J}^{\circ})
\]

\[
    x \in \widetilde{I}^{\circ} \subset \widetilde{I}'^{\circ} \cap \widetilde{I}''^{\circ}
  \]

\[
  \mathcal{L}' = \bigcup_{I \in S_0} \widetilde{I}, \qquad S' = \lbrace I \in S \mid \widetilde{I} \subset \mathcal{L}' \rbrace
\]

\[
  \begin{array}{ll}
    S' \to \mathfrak{P}(\mathcal{L}') & I \mapsto \widetilde{I} \\
    S' \to \mathfrak{P}_2(\mathcal{L}') & I \mapsto \partial I
  \end{array}
\]

\[
  \begin{array}{ll}
    S' \to \mathfrak{P}(\mathcal{L}'), & I \mapsto \widetilde{I} \\
    S' \to \mathfrak{P}_2(\mathcal{L}'), & I \mapsto \partial I .
  \end{array}
\]

\[
  [x_0,x] = \widetilde{I}_{x_0,x}, \qquad [y,x_1] = \widetilde{I}_{x_1,y}.
\]

\[
  x \preccurlyeq y,\ y \preccurlyeq z \Longrightarrow x \preccurlyeq z
\]

\[
  y \in [x,x_1] \text{ et } z \in [y,x_1] \Longrightarrow z \in [x,x_1]
\]

\[
  [x,x_1] \overset{\text{déf}}{=}
  \left\lbrace
  \begin{array}{ll}
    \widetilde{I}_{x_1,x} & \text{si } x \neq x_0, x_1 \\
    \widetilde{I} & \text{si } x = x_0 \\
    \{x_1\} & \text{si } x = x_1
  \end{array}
  \right.
\]

\[
  [x_0,x] = \left\lbrace
  \begin{array}{ll}
    \{x_0\} & \text{si } x = x_0 \\
    \mathcal{L} = \widetilde{I} & \text{si } x = x_1 \\
    \widetilde{I}_{x_0,x} & \text{si } x \in \widetilde{I}^{\circ} = \mathcal{L} \smallsetminus \{x_0,x_1\}
  \end{array}
  \right.
\]

\[
  [x,x_1] = \lbrace \cdots
\]

\[
  x \preccurlyeq y
\]

\[
  \widetilde{I}_{x,y} = \lbrace z \in \mathcal{L} \mid x \leq z \leq y \rbrace, \qquad \partial I_{x,y} = \{x,y\}
\]

\[
  \exists\, I \in S \text{ tel que } x, y \in \widetilde{I}
\]

\[
  \exists\, I \in S \text{ tel que } x, y \in \widetilde{I}^{\circ}\ ].
\]

\[
  \coprod_{I \in S} \partial I \;\amalg\; \coprod_{x \in \mathcal{L}} B_x
\]

\[
  (I,x) \underset{(a)}{\sim} (J,y) \Longleftrightarrow J \subset I
\]

\[
  (I,x) \underset{(b)}{\sim} (x, \beta(I,x))
\]

\[
  x \prec y \overset{\text{déf}}{\Longleftrightarrow} \bigl(\exists\, I \in S \text{ tel que } x \in \operatorname{or}(I),\ y \in \operatorname{ex}(I)\bigr)
\]

\[
  x_i \geqslant y_i \leqslant x_{i+1} \geqslant y_{i+1} \leqslant x_{i+2}
\]

\[
  x \underset{n-1}{\sim} z, \qquad z \underset{1}{\sim} y .
\]

\[
  \mathcal{L} = \underbrace{\bigl(\mathbb{Q} \cap {]{-\infty}, r]}\bigr)}_{L_1} \amalg \underbrace{\bigl(\mathbb{Q} \cap [r, +\infty]\bigr)}_{L_2} \times E
\]

\[
  \mathcal{L} \xrightarrow{\;p\;} \mathbb{Q} \quad (\text{surjectif}), \qquad
  \mathcal{L} \xrightarrow{\;q\;} E \amalg \{0\}, \quad q(x) = 0 \text{ si } x \in \mathcal{L}
\]

\[
  x < y \Longleftrightarrow px < py \ \underline{\text{et}}
\]

\[
  x < y \ \text{ssi} \ \left|\ \begin{array}{l}
    \text{ou bien } x, y \text{ dans un même } \mathcal{L}_i, \text{ et } x < y \text{ dans cet } \mathcal{L}_i \\
    \text{ou bien } x \in \mathcal{L}_0,\ y \in \mathcal{L}_i \ (i \in I)
  \end{array}\right.
\]

\[
  I_{x,u} \cap I_{x,v} = \{x\}.
\]

\[
  \operatorname{br}_t I_{x,t} \neq \operatorname{br}_t J_{x,t},
\]