\[
  \mathcal{M},\ \underset{\text{incidence immédiate}}{\lhd},\ \mathring{\ll},\ |\circ|,\
  \underset{\text{involution sans pt fixe}}{X \mapsto X^{-}},\ \mathcal{M}_0^{+}
\]

\[
  \leq,\quad \ll,\quad |\circ|
\]

\[
  X \leq Y \Longrightarrow X \ll Y .
\]

\[
  \widetilde{Y}^{X} = \{ Z \in \widetilde{Y} \mid X \ll Z \}
\]

\[
  X |\circ| Y,\ X' \mathring{\ll} X,\ Y' \mathring{\ll} Y \Longrightarrow X' |\circ| Y' .
\]

\[
\begin{aligned}
  X \lessgtr Y &\overset{\text{déf}}{\Longleftrightarrow} \forall X' \in \widetilde{X} \smallsetminus \widetilde{X} \cap \widetilde{Y},\ Y' \in \widetilde{Y} \smallsetminus \widetilde{X} \cap \widetilde{Y},\ \text{on a } X' |\circ| Y' \\
  X \parallel Y &\overset{\text{déf}}{\Longleftrightarrow} \forall X' \in \widetilde{X},\ Y' \in \widetilde{Y},\ \text{on a } X' |\circ| Y' .
\end{aligned}
\]

\[
  X \parallel Y \Longleftrightarrow X \lessgtr Y,\ \text{et } \widetilde{X} \cap \widetilde{Y} = \emptyset
\]

\[
  \mathcal{F}_{\mathcal{M}} = \{ \mathfrak{F} \subset \mathcal{M} \mid \mathfrak{F} \text{ fermé dans } \mathcal{M} \text{ pour } \leq, \text{ et } \forall X, Y \in \mathfrak{F},\ \text{on a } X \lessgtr Y \}
\]

\[
  \mathcal{M} \hookrightarrow \mathcal{F}_{\mathcal{M}},\qquad X \mapsto \widetilde{X} = \mathcal{M}_{\leq X}
\]

\[
  F \leq G \overset{\text{déf}}{\Longleftrightarrow} F \subset G
\]

\[
\begin{aligned}
  F \ll G &\overset{\text{déf}}{\Longleftrightarrow} \forall X \in \widetilde{F},\ \exists\, Y \in \widetilde{G},\ \text{avec } X \ll Y \\
  F \lessgtr G &\Longleftrightarrow \forall X \in \widetilde{F},\ Y \in \widetilde{G},\ \text{on a } X \lessgtr Y \\
  &\Longleftrightarrow \{F, G\} \text{ majoré dans } \mathcal{F} \\
  &\Longleftrightarrow \mathrm{Sup}(F,G) \text{ existe dans } \mathcal{F} \\
  &\Longleftrightarrow F \cup G \in \mathcal{F}
\end{aligned}
\]

\[
  \text{\struck{$Z \ll Y \Longleftrightarrow X \ll Y$.}}
\]

\[
  Z \ll Y \Longleftrightarrow \text{\struck{$Z$}}\, X \ll Y
\]

\[
  Z \mathring{\ll} X,\ Z \ll Y \Longrightarrow X \ll Y \qquad (\text{si } X \lessgtr Y).
\]

\[
  \text{\struck{$Z \mathring{\ll} X,\ Z \mathring{\ll} Y,\ X \lessgtr Y \Longrightarrow X = Y$.}}
\]

\[
  X \mathring{\ll} Y \Longleftrightarrow X \leq Y
\]

\[
\begin{aligned}
  F \parallel G &\Longleftrightarrow \forall X \in \widetilde{F},\ Y \in \widetilde{G},\ \text{on a } X \parallel Y \\
  &\Longleftrightarrow F \lessgtr G \text{ et } F \cap G = \emptyset
\end{aligned}
\]

\[
  \text{alors}\quad G \lessgtr F \;\text{\struck{$\Longleftrightarrow$}}\; \forall i,\ G \lessgtr F_i .
\]

\[
  \begin{array}{ccc}
    G & \ll & F \\
      &     & \geq \\
      &     & F'
  \end{array}
\]

\[
  \begin{array}{ccc}
    G  & \ll & F \\
    \geq &   & \geq \\
    G' & \ll & F'
  \end{array}
\]

\[
  \widetilde{G'} = \widetilde{G} \cap \mathrm{Omb}(F')
\]

\[
  F' \longmapsto G_{F'} : \mathrm{Ssfig}(F) \longrightarrow \mathrm{Ssfig}(G)
\]

\[
  \mathrm{Inf}^{\ll}(G, F') = \mathrm{Inf}^{\leq}(G, F')
\]

\[
\begin{aligned}
  \text{a)}\quad & F' \lessgtr G' \Longleftrightarrow F'_{L} \lessgtr G'_{L} \\
  \text{b)}\quad & F' \parallel G' \Longleftrightarrow F'_{L} \parallel G'_{L}
\end{aligned}
\]

\[
  F \parallel G \Longrightarrow F' \parallel G'
\]

\[
\begin{array}{ccccc}
  \mathrm{Raff}(F) & \longrightarrow & \prod_i \mathrm{Raff}(F_i) & \rightrightarrows & \prod_{i,j} \mathrm{Raff}(F_i \cap F_j) \\
  G & \longmapsto & (G|F_i)_i & &
\end{array}
\]

\[
  \mathrm{Raff}(F) \longrightarrow \prod_{X \in \widetilde{F}} \mathrm{Raff}(X)
\]

\[
  \mathrm{Raff}\, F = \varprojlim_{X \in \widetilde{F}} \mathrm{Raff}(X)
\]

\[
  \underset{\text{ens.\ des lieux}}{L \subset \mathcal{M}},
\]

\[
  x \parallel y \Longleftrightarrow x |\circ| y .
\]

\[
\begin{aligned}
  \mathrm{omb}(X) &= \{ x \in L \mid x \ll X \} = \mathrm{Omb}(X) \cap L \\
  \mathrm{omb}(F) &= \textstyle\bigcup_{X \in F} \mathrm{omb}(X) \qquad \text{pour } F \subset \mathcal{M} \text{ (p.ex.\ } F \in \mathcal{F}) \\
  \text{\struck{$\mathrm{omb}(X)^{\circ}$}} &\;\text{\struck{$= \mathrm{omb}(X)$}} \\
  \mathrm{omb}(X)^{\circ} &= \mathrm{omb}(X) - \mathrm{omb}(\partial X)
\end{aligned}
\]

\[
  \partial X = \widetilde{X} \smallsetminus \{X\} \;\text{\struck{$F$}}\; = \{ Y \in \mathcal{M} \mid Y \lneq X \} \in \mathcal{F}
\]

\[
  x \in \mathrm{omb}(X)^{\circ} \Longleftrightarrow x \mathring{\ll} X
\]

\[
  X \parallel Y \Longrightarrow \forall x \in \mathrm{omb}(X)^{\circ},\ y \in \mathrm{omb}(Y)^{\circ},\ \text{on a } x \parallel y .
\]

\[
\begin{aligned}
  \mathcal{M} &\longrightarrow \mathrm{Figél}(L) &\quad&\text{ou}\quad& \mathcal{F} &\longrightarrow \mathrm{Fig}(L) \\
  X &\longmapsto \mathrm{mulomb}(X) &&\text{ou}& F &\longmapsto \mathrm{mulomb}(F)
\end{aligned}
\]

\[
  \forall x \in \mathrm{omb}(X)^{\circ},\ y \in \mathrm{omb}(Y)^{\circ},\ \text{on a } x \parallel y .
\]

\[
  x \parallel y \quad \text{relation \emph{sur $L$}}
\]

\[
\begin{aligned}
  \mathrm{Ssfig}(F) &\xrightarrow{\;\sim\;} \mathrm{Ssfig}(\mathrm{mulomb}(F)) \\
  F' &\longmapsto \mathrm{mulomb}(F')
\end{aligned}
\]

\[
\begin{aligned}
  \text{b)}\quad & \mathrm{mulomb}(F) \ll \mathrm{mulomb}\, G \Longrightarrow F \ll G \\
  \text{c)}\quad & \mathrm{mulomb}(F) \lessgtr \mathrm{mulomb}(G) \Longrightarrow F \lessgtr G .
\end{aligned}
\]

\[
  \mathcal{M} \subset \mathrm{Figél}(L),
\]

\[
  \widetilde{F}_{\leq X} = ( Y \in F \mid Y \subset X ) \subset F \text{ est aussi dans } \mathcal{M}
\]

\[
  \widetilde{X} \quad (X \in \mathcal{M})
\]

\[
  X = \mathrm{Sup}\, F_i \Longrightarrow \exists\, i,\ F_i = X .
\]

\[
  \widetilde{F} = \{ X \in \mathcal{M} \mid X \lhd F \}
  \qquad \text{On pose } F \leq G \overset{\mathrm{def}}{\Longleftrightarrow}
  \widetilde{F} \subset \widetilde{G}
\]

\[
  X \lhd F \Longleftrightarrow F_{X} \leq F \qquad
  (\Longleftrightarrow X \in \widetilde{F})
\]

\[
  \mathcal{M} \longrightarrow \mathfrak{F} \qquad X \longmapsto F_{X}
  \quad \text{est injectif.}
\]

\begin{align*}
  F_{X} \leq F_{Y}
  &\Longleftrightarrow \forall\, \text{\struck{$\mathfrak{L} \in \mathfrak{F}$}}\
    F \in \mathfrak{F},\ Y \lhd F \Rightarrow X \lhd F \\
  &\Longleftrightarrow \forall\, \mathfrak{L} \in \widetilde{\mathfrak{F}},\
    Y \in \mathfrak{L} \Rightarrow X \in \mathfrak{L} \\
  &\Longleftrightarrow X \lhd F_{Y} \Longleftrightarrow X \in \widetilde{F}_{Y}
\end{align*}

\[
  X \leq Y ,
\]

\begin{align*}
  F \lessgtr G &\overset{\mathrm{def}}{\Longleftrightarrow}
    \{F, G\} \text{ majoré dans } \mathfrak{F} \Longleftrightarrow
    F \cup G \text{ \uncertain{existe}} \\
  &\Longleftrightarrow \widetilde{F} \cup \widetilde{G} \in \widetilde{\mathfrak{F}} \\
  X \lessgtr Y &\overset{\mathrm{def}}{\Longleftrightarrow} F_{X} \lessgtr F_{Y}
    \text{ i.e. } \widetilde{X} \cup \widetilde{Y} \in \widetilde{\mathfrak{F}}
    \quad [\widetilde{X} = \mathcal{M}_{\leq X}]
\end{align*}

\[
  F \lessgtr G \Longleftrightarrow \forall X \in \widetilde{F},\
  Y \in \widetilde{G}, \text{ on a } X \lessgtr Y
\]

\[
  X \lessgtr Y \Longleftrightarrow \forall X' \in \widetilde{X} \setminus
  \widetilde{X} \cap \widetilde{Y},\
  Y' \in \widetilde{Y} \setminus \widetilde{X} \cap \widetilde{Y},
\]

\[
  X \ll_{\mathrm{pol}} Y \Longleftrightarrow X \ll Y, \text{ et }
  \forall Y' \in \widetilde{Y},\ X_{Y'} \overset{\mathrm{def}}{=}
  \widetilde{X} \cap \mathrm{Ombr}(Y')
\]

\[
  X \leq Y \Longrightarrow X \ll_{\mathrm{pol}} Y
\]

\[
  (\mathcal{M}, \leq, \ll_{\mathrm{pol}}, |\circ|)
\]

\[
  X \ll_{\mathrm{pol}} Y \ll_{\mathrm{pol}} Z \Longrightarrow X \ll_{\mathrm{pol}} Z .
\]

\[
  \widetilde{Y}^{X,\mathrm{pol}} = \{ Z \in \widetilde{Y} \mid X \ll_{\mathrm{pol}} Z \}
\]

\[
  \widetilde{Y}^{X,\mathrm{pol}} = \widetilde{Y}^{X} ,
\]

\[
  X \ll Y' \leq Y \Longrightarrow X \ll_{\mathrm{pol}} Y' ,
\]

\[
  X \mathrel{\mathring{\ll}_{\mathrm{pol}}} Y \Longleftrightarrow
  X \ll_{\mathrm{pol}} Y \text{ et } X \mathrel{\mathring{\ll}} Y .
\]

\[
  \mathcal{M}^{\mathrm{pol}} = (\mathcal{M}, \leq, \ll_{\mathrm{pol}}, |\circ|)
\]

\[
  X \lessgtr_{\mathrm{pol}} Y \overset{\mathrm{def}}{\Longleftrightarrow}
  X \lessgtr Y, \text{ et } \forall X' \in \widetilde{X},\ Y' \in \widetilde{Y},
\]

\[
  \mathrm{Figpol}(\mathcal{M}) \subset \mathrm{Fig}(\mathcal{M})
\]

\[
  \left\{
  \begin{array}{l}
    1^{\circ})\ \forall X' \in \widetilde{X} \setminus \widetilde{X} \cap \widetilde{Y},\
      Y' \in \widetilde{Y} \setminus \widetilde{X} \cap \widetilde{Y}
      \Longrightarrow X' \mathrel{|\circ|} Y' \\
    2^{\circ})\ \forall X' \in \widetilde{X},\ Y' \in \widetilde{Y},\
      \widetilde{X}' \cap \widetilde{Y}' \text{ est vide ou a un plus grand élément.}
  \end{array}
  \right.
\]

\[
  \mathfrak{F} \subset \mathfrak{P}(\mathcal{M}), \text{ ou plutôt }
  \mathfrak{F} \subset \mathrm{Figpol}(\mathcal{M}),
\]

\[
  X \lessgtr Y \Longrightarrow X \lessgtr_{\mathrm{pol}} Y
\]

\[
  (*) \qquad X \ll Y \Longleftrightarrow \exists Y',\
  X \mathrel{\mathring{\ll}} Y' \leq Y .
\]

\[
  (\mathcal{M}, \leq, \mathring{\ll}, |\circ|) ,
\]

\[
  (2) \qquad X \leq Y \mathrel{\mathring{\ll}} Z \Longrightarrow
  \text{\struck{$\exists$}}\ \exists Z' \text{ avec } X \mathrel{\mathring{\ll}} Z' \leq Z ,
\]

\[
  (X \leq Y \text{ et } X \mathrel{\mathring{\ll}} Y) \Longleftrightarrow X = Y ,
\]

\[
  \leq,\ \mathring{\ll},\ |\circ|
\]

\[
  (\mathcal{M}, \mathfrak{F} \mid \lhd, \mathring{\ll}, |\circ|)
\]

\[
  \mathcal{L} = \{ X \in \mathcal{M} \mid X \text{ minimal pour } \leq
  \text{ et pour } \mathring{\ll} \} .
\]

\[
  (\mathcal{M}, \leq, \mathring{\ll}, \|) \qquad \text{avec}
\]

\[
  X \mathrel{|\circ|} Y \Longleftrightarrow X^{\circ} \cap Y^{\circ} = \emptyset
\]

\[
  X \mathrel{\mathring{\ll}} Y \overset{\mathrm{def}}{\Longleftrightarrow} X = Y ,
  \qquad X \mathrel{|\circ|} Y \overset{\mathrm{def}}{\Longleftrightarrow} X \neq Y ,
\]

\[
  \begin{array}{lll}
    X \ll Y & \Longleftrightarrow & X \leq Y \\
    X \lessgtr Y & & \text{toujours satisfait} \\
    X \| Y & \text{ssi} & \widetilde{X} \cap \widetilde{Y} = \emptyset
      \text{ i.e. } \{X, Y\} \text{ pas minoré.}
  \end{array}
\]

\[
  \begin{array}{l}
    \mathrm{omb}(X) = \{ x \in \mathcal{L} \mid x \leq X \} \\
    \mathrm{omb}(X)^{\circ} = \text{\struck{$\{ x \in \mathcal{L} \mid x < X \}$}}
  \end{array}
  \left\{
  \begin{array}{ll}
    \emptyset & \text{si } X \notin \mathcal{L} \\
    \mathrm{omb}(X) = \{X\} & \text{si } X \in \mathcal{L}
  \end{array}
  \right.
\]

\[
  I \xrightarrow{\ \sim\ } \widetilde{F}, \qquad i \longmapsto X_{i}
\]

\[
  \varphi : I \longrightarrow \mathcal{M}
\]

\[
  \widetilde{F} = \varphi(I) .
\]

\[
      \widetilde{i} = I_{\leq i} \xrightarrow{\ \sim\ } \widetilde{X}_{i} =
      \mathcal{M}_{\leq X_{i}}
    \]

\[
  X' \in \widetilde{X}_{i} \setminus \widetilde{X}_{i} \cap \widetilde{X}_{j},\
  Y' \in \widetilde{X}_{j} \setminus \widetilde{X}_{i} \cap \widetilde{X}_{j}
  \Longrightarrow X' \mathrel{|\circ|} Y' .
\]

\[
  X \ll Y \Longrightarrow X' \ll Y'
\]

\[
  X \lessgtr Y \overset{?}{\Longrightarrow} X' \lessgtr Y' , \qquad
  X \| Y \overset{?}{\Longrightarrow} X' \lessgtr Y'
\]

\[
      \widetilde{X} \xrightarrow{\ \sim\ } \widetilde{X}' \qquad \text{où }
      X' = \varphi(X) .
    \]

\[
  X \lessgtr Y \Longrightarrow X' \lessgtr Y' , \qquad
  X \| Y \Longrightarrow X' \| Y' .
\]

\[
  \begin{array}{l}
    \text{\struck{$X \lessgtr Y \Longleftrightarrow X' \lessgtr Y'$}} \\
    \text{\struck{$X \| Y$}} \\
    X \leq Y \Longleftrightarrow X' \leq Y' \\
    \text{\struck{$X \ill{}$}} \\
    X \| Y \Longleftrightarrow X' \| Y'
  \end{array}
\]

\[
  F' \lessgtr G' , \quad \varphi(F \cup G) = \varphi(F) \cup \varphi(G) ,
  \quad \varphi(F \cap G) = \varphi(F) \cap \varphi(G)
\]

\[
  \mathcal{M} \longrightarrow \text{Figél}(\mathcal{M}) \qquad
  X \longmapsto \mathrm{Multombs}(X)
\]

\[
  \begin{array}{l}
    |\mathrm{Multombs}(X)| = \mathrm{Omb}(X) = \mathcal{M}_{\ll X} \\
    |\mathrm{Multombs}(X)|^{\circ} = \mathrm{Omb}^{\circ}(X) =
      \mathcal{M}_{\mathring{\ll} X}
  \end{array}
\]

\[
  \text{\struck{$\mathrm{Multombs}(X) \mathrel{|\circ|} \mathrm{Multombs}(Y) \Longleftrightarrow$}}
\]

\[
  \begin{array}{rl}
    \mathrm{Multombs}(X) \mathrel{|\circ|} \mathrm{Multombs}(Y)
    & \Longleftrightarrow \mathrm{Omb}^{\circ}(X) \cap \mathrm{Omb}^{\circ}(Y) = \emptyset \\
    & \Longleftrightarrow \nexists Z \in \mathcal{M}, \text{ avec }
      Z \mathrel{\mathring{\ll}} X,\ Z \mathrel{\mathring{\ll}} Y .
  \end{array}
\]

\[
  X \mathrel{|\circ|} Y \Longrightarrow \mathrm{Multombs}(X) \mathrel{|\circ|}
  \mathrm{Multombs}(Y)
\]

\[
  (*) \qquad X \mathrel{|\circ|} Y \Longleftrightarrow \{X, Y\}
  \text{ non minoré dans } \mathcal{M}_{\mathring{\ll}}
\]

\[
  \begin{array}{lll}
    \text{Magasins généraux} & \leq,\ \mathring{\ll},\ |\circ| & \text{Mag 1 -- Mag 4} \\
    \text{Magasins} & \leq,\ \mathring{\ll} & \text{Mag 1', Mag 2' = Mag 2}
      \ (\text{p.~22}) \ (\text{\uncertain{cette} page}) \\
    \text{Magasins quasi-ensemblistes} & & \text{Mag 1', Mag 2', Mag 3'} \ (\text{p.~24}) \\
    \text{Magasins ensemblistes} & & \text{Magens 1, Magens 2} \ (\text{p.~9})
  \end{array}
\]

\[
  \varphi : \mathcal{M} \longrightarrow \text{Figél}(\mathcal{L}) \qquad
  X \longmapsto \mathrm{multombs}(X)
\]

\[
  \begin{array}{l}
    \text{\struck{$X \mathrel{|\circ|} Y \Longrightarrow \mathrm{omb}(X)$}} \\
    |\mathrm{multombs}(X)| = \mathrm{omb}(X) \\
    |\mathrm{multombs}(X)|^{\circ} = \mathrm{omb}(X)^{\circ}
  \end{array}
\]

\[
  \mathrm{multombs}(X) \mathrel{|\circ|} \mathrm{multombs}(Y)
  \Longleftrightarrow \mathrm{omb}(X)^{\circ} \cap \mathrm{omb}(Y)^{\circ} = \emptyset
\]

\[
  X \mathrel{|\circ|} Y \Longrightarrow \mathrm{multombs}(X) \mathrel{|\circ|}
  \mathrm{multombs}(Y) .
\]

\[
  \text{Figél}(E) \longrightarrow \text{Figél}(E')
\]

\[
X \longmapsto |X| \qquad M \longrightarrow \mathfrak{P}(P),
\]

\[
|Y| \subset |Z| \Longrightarrow Y \leq Z \quad ??
\]

\[
|Y|^\circ \overset{?}{=} |Y| \smallsetminus \bigcup_{\substack{Z \in \tilde{X} \text{ tel que}\\ |Z| \subsetneq |Y|}} |Z| \overset{\text{déf}}{=} |Y|^{\circ\prime}
\]

\[
|X| = \bigcup_{Y \in \tilde{X}} |Y|^\circ .
\]

\[
|Y| \subset |Z| \Longrightarrow Y \leq Z ,
\]

\[
\text{\struck{$|Y| = \coprod_{Z \in \tilde{Y}} |Z|^\circ = \bigcup_{Z \in L}$}}
\]

\[
|Y| = \bigcup_{Z \in \tilde{Y}} |Z|^\circ = \Big(\bigcup_{Z \in L} |Z|^\circ\Big) \cup \Big(\bigcup_{Z \in \tilde{Y} \smallsetminus L} |Z|^\circ\Big)
\]

\[
|Z| = \Big(\bigcup_{Z \in L} |Z|^\circ\Big) \cup \Big(\bigcup_{Z \in \tilde{Z} \smallsetminus L} |T|^\circ\Big)
\]

\[
|Y| \cap |Z| = \bigcup_{Z \in L} |Z|^\circ ,
\]

\[
x \in |Y|^\circ,\ x \in |Z| \Longrightarrow Y \leq Z .
\]

\[
\text{\struck{$\varphi : I \longrightarrow \mathfrak{P}(P), \quad i \longmapsto X_i \text{ ou } |i|$}}
\]

\begin{align*}
\partial X_i &= \bigcup_{j<i} X_j \;\subset X_i \\
X_i^\circ &= X_i \smallsetminus \partial X_i \\
|\Phi| &= \bigcup_{i \in I} X_i .
\end{align*}

\[
I^x = \{ i \in I \mid x \in X_i \}
\]

\[
x \in X_i^\circ \Longleftrightarrow i \text{ est un élément minimal de } I^x .
\]

\[
  X_i \cap X_j = \bigcup_{k \leq i, j} X_k .
  \]

\[
  X_i = \bigcup_{j \leq i} X_j^\circ \qquad \text{(automat. si $I$ fini)}
  \]

\[
  I' = \{ i \in I \mid X_i^\circ \neq \emptyset \} ,
  \]

\[
  X_j^\circ \subset X_i \Longrightarrow j \leq i .
  \]

\[
  x \in X_i^\circ .
  \]

\[
  \forall i, j \in I,\ x \in X_i^\circ \cap X_j^\circ \Longrightarrow i = j
  \]

\[
  \forall i \in I \text{ tel que } x \in X_i,\ \exists\, j \leq i \text{ tel que } x \in X_j^\circ .
  \]

\[
\text{(i)} \Longleftrightarrow \text{(iii)} .
\]

\[
  I'(i) = \{ j \in I' \mid X_j^\circ \subset X_i \} ,
  \]

\[
  \text{\struck{$I(i) = \{ j \in I \mid X_j^\circ \subset X_i \}$}}
  \]

\[
  X_i = \bigcup_{j \in I'(i)} X_j^\circ
  \]

\[
  I'(i) = I' \cap I_{\leq i}
  \]

\[
\left\{
\begin{array}{lll}
|\Phi| = \bigcup_{i \in I} X_i^\circ & & \text{i.e. (1')} \\
\forall i, j \in I & X_i^\circ \cap X_j^\circ = \emptyset & \text{i.e. (2')} \\
\forall i \in I & X_i = \bigcup_{j \leq i} X_j^\circ & \text{i.e. (3')} \\
\end{array}
\right.
\]

\[
\forall i, j \in I \quad X_j^\circ \subset X_i \Longrightarrow j \leq i .
\qquad 3'') \Leftrightarrow \gamma)
\]

\[
\text{(iv)} \Longleftrightarrow \text{(i)} + \gamma)
\]

\[
(j \leq i) \Longleftrightarrow (X_j \subset X_i) \Longleftrightarrow (X_j^\circ \subset X_i) \Longleftrightarrow (X_j^\circ \cap X_i \neq \emptyset)
\]

\[
X_i = \bigcup_{i' \leq i} X_{i'}^\circ , \qquad X_j = \bigcup_{j' \leq j} X_{j'}^\circ
\]

\[
i \leq j \Longrightarrow X_i \subset X_j \Longrightarrow \text{\struck{\ill{}}}
\quad \text{(si on suppose $X_j^\circ \neq \emptyset$)}
\]

\[
(X_j \subset X_i) \Longrightarrow (X_j^\circ \subset X_i) \overset{\text{si } X_j^\circ \neq \emptyset}{\Longrightarrow} (X_j^\circ \cap X_i)
\]

\[
\text{\struck{$X_j^\circ \cap X_i \Longrightarrow i \leq j$}}
\]

\[
X_j^\circ \cap X_i \neq \emptyset \Longrightarrow j \leq i .
\]

\[
\text{(i)} \Longleftrightarrow \text{(iii)} \Longleftrightarrow \text{(iv)}
\]

\[
i \longmapsto X_i : \varphi : I \longrightarrow \mathfrak{P}(X)
\]

\[
X_i \subset X_j \Longleftrightarrow i \leq j .
\]

\[
X_i \cap X_j = \bigcup_{k \leq i, j} X_k
\]

\[
X_i = \bigcup_{i' \leq i} X_{i'}^\circ , \qquad X_j = \bigcup_{j' \leq j} X_{j'}^\circ
\]

\[
X_i \cap X_j = \bigcup_{\substack{i' \leq i \\ j' \leq j}} X_{i'}^\circ \cap X_{j'}^\circ .
\]

\[
X_i \cap X_j = \bigcup_{k \leq i, j} X_k^\circ \subset \bigcup_{k \leq i, j} X_k
\qquad \text{cqfd}
\]

\[
X_i \cap X_j = \bigcup_{k \leq i, j} X_k .
\]

\[
X_i^\circ \cap X_j^\circ \subset X_i \cap X_j = \bigcup_{k \leq i, j} X_k
\]

\[
i \neq j \Longrightarrow X_i \cap X_j \subset \partial X_i \cup \partial X_j \subset \complement (X_i^\circ \cap X_j^\circ)
\]

\[
X_j^\circ \cap X_i \neq \emptyset \Longrightarrow j \leq i .
\]

\[
X_j^\circ \cap X_i \subset X_j \cap X_i = \bigcup_{k \leq i, j} X_k
\]

\[
X_j^{\circ\prime} = \text{cellule ouverte de } X_j \text{ dans } \Phi_i
= X_j \smallsetminus \bigcup_{\substack{j' \leq i \\ \text{t.q. } X_{j'} \subsetneq X_j}} X_{j'}
\]

\[
X_j^\circ = X_j \smallsetminus \bigcup_{j' < j} X_{j'} ,
\]

\[
j' < j \Longrightarrow X_{j'} \subsetneq X_j .
\]

\[
X_j^{\circ\prime} \subset X_j^\circ
\]

\[
\bigcup_{j \leq i} X_j^{\circ\prime} \subset \bigcup_{j \leq i} X_j^\circ \subset X_i
\]

\[
X_j^{\circ\prime} = X_j^\circ :
\]

\[
\text{\struck{$A_\infty$}}\ N = \bigcap A_n \neq \emptyset .
\]

\[
A_i^\circ = A_i \smallsetminus A_{i+1} \neq \emptyset \quad \text{si } i \neq \infty
\]

\[
A_\infty^\circ = A_\infty \neq \emptyset .
\]

\[
\left\{
\begin{array}{l}
\forall i \quad X_i^\circ \neq \emptyset \\
\forall i \neq j \quad X_i^\circ \cap X_j^\circ = \emptyset \\
\text{\struck{$\bigcup X_i^\circ = |\Phi|$}}
\end{array}
\right.
\]

\[
X_i = \bigcup_{j \text{ t.q. } X_j^\circ \subset X_i} X_j
\]

\[
X_j^\circ \subset X_i \Longrightarrow j \leq i
\]

\[
X_j \subset X_i \Longrightarrow j \leq i
\]

\[
X_j^\circ \subset X_i \Longrightarrow \text{\struck{$j \leq$}}\ X_j \subset X_i \ ?
\]

\[
A = A_\infty - N
\]

\[
\Phi = \{ (A_i), A_\infty, A \}
\]

\[
A_i^\circ = A_i - A_{i-1} , \qquad A_\infty^\circ = N , \qquad A^\circ = A
\]

\[
f^* : \mathrm{Figures}(\mathcal{L}) \longrightarrow \mathrm{Figures}(\mathcal{L}') .
\]

\[
\text{(1)} \qquad f^*(\Phi) = \{ f^{-1}(A) \mid A \in \Phi,\ f^{-1}(A^\circ) \neq \emptyset \}
\]

\[
\text{i.e. } A^\circ \cap \mathcal{L}_0 \neq \emptyset, \quad \text{où } \mathcal{L}_0 = f(\mathcal{L}')
\]

\[
A \longmapsto \text{\struck{$A$}}\ F_{\mathcal{L}_0} \cdot A
\]

\[
\mathrm{Figures}(\mathcal{L}) \longrightarrow \mathrm{Figures}(\mathcal{L})_{\mathcal{L}_0} ,
\]

\[
\mathrm{Figures}_{\mathcal{L}_0}(\mathcal{L}) \xrightarrow{\ \sim\ } \mathrm{Figures}(\mathcal{L}_0)
\]

\[
f_0^* : \mathrm{Figures}(\mathcal{L}_0) \longrightarrow \mathrm{Figures}(\mathcal{L}') ,
\]

\[
\text{(2)} \qquad f_0^*(\Phi_0) = \{ f^{-1}(A) \mid A \in \Phi_0 \} ,
\]

\[
X^\circ \cap \mathcal{L}_0 = \emptyset \quad \text{i.e.} \quad F_{\mathcal{L}_0} \mathbin{|{\circ}|} X
\]

\[
\mathrm{Figures}_{\mathcal{L}_0}(\mathcal{L}_0) \xrightarrow{\ \sim\ } \mathrm{Figures}(\mathcal{L})_{\mathcal{L}_0} \hookrightarrow \mathrm{Figures}(\mathcal{L}) ,
\]

\[
i_* : \mathrm{Figures}(\mathcal{L}_0) \longrightarrow \mathrm{Figures}(\mathcal{L})
\]

\[
i^* i_* = \mathrm{id}_{\mathcal{F}_0} , \qquad
\text{\struck{$i_* i^*$}}\ i_* i^*(X) = i_*(\mathbb{1}) \cdot X , \qquad \text{plus généralement}
\]

\[
i_*(X') \cdot Y = i_*(X' \cdot i^*(Y))
\qquad \text{(formule de projection)}
\]

\[
X \mathring{\ll} Y \Longrightarrow X' \mathring{\ll} Y'
\]

\[
X \leq Y \Longrightarrow X' \leq Y' \quad \text{ou seulement } X' \ll Y'
\]

\[
X \mathring{\ll} Y \Longrightarrow X' \mathring{\ll} Y' \quad \text{mais aussi}
\]

\[
X \leq Y \Longrightarrow X' \leq Y' \quad \text{donc aussi}
\]

\[
X \ll Y \Longrightarrow X' \ll Y'
\]

\[
X \mathbin{|{\circ}|} Y \Longrightarrow X' \mathbin{|{\circ}|} Y' .
\]

\[
\mathrm{Fig}(\mathcal{L}) \longrightarrow \mathrm{Fig}(\mathcal{L} \times \mathcal{L}')
\]

\[
F \longmapsto F \times F' = \mathrm{pr}_1^*(F) \cdot \mathrm{pr}_2^*(F')
\]

\[
\Phi \longmapsto \Phi \cdot F \qquad (\text{où } F = \mathrm{pr}_2^*(F')) .
\]

\[
\leq ,\quad \mathring{\ll} ,\quad \mathbin{|{\circ}|} \quad /\!/ \quad \text{\struck{$\ll$}} ,\quad \|
\]

\[
\|_{M'} = \|_M \mid M' , \qquad \ll_{M'} = \ll_M \mid M .
\]

\[
  \mathcal{F}' = \{ F \in \mathcal{F} \mid \widetilde{F} \subset \mathcal{M}' \}.
\]

\[
  (\mathcal{F}, \mathcal{M}, \lhd, \mathrel{\mathring{\ll}}, |\circ|), \qquad
  (\mathcal{F}', \mathcal{M}', \lhd, \mathrel{\mathring{\ll}}, |\circ|)
\]

\[
  f : \mathcal{F} \longrightarrow \mathcal{F}'
\]

\[
  f_{\mathcal{M}} : \mathcal{M} \longrightarrow \mathcal{F}',
\]

\[
  (*) \qquad f(F \cap G) = f(F) \cap f(G).
\]

\[
  (*)_{\mathcal{M}} \qquad f_{\mathcal{M}}(X) \cap f_{\mathcal{M}}(Y) = \operatorname*{Sup}_{Z \in \widetilde{X} \cap \widetilde{Y}} f_{\mathcal{M}}(Z)
\]

\[
  X_i \cap X_j = \bigcup_{k \leq i, j} X_k \ ).
\]

\[
  (**) \qquad \forall F \in \mathcal{F}, \text{ posant } F' = f(F), \text{ l'application}
\]

\[
  G \longmapsto G' = f(G) : \operatorname{SsFig}(F) \xrightarrow{\ \sim\ } \operatorname{SsFig} F'
\]

\[
  f_{\mathcal{M}} = f : \mathcal{M} \longrightarrow \mathcal{M}'
\]

\[
  f \,|\, \widetilde{F} : \widetilde{F} \xrightarrow{\ \sim\ } \widetilde{F}' \qquad \text{isom.\ d'ens.\ ordonnés}
\]

\[
  \mathcal{M} \longrightarrow \operatorname{Figures}(\mathcal{L}).
\]

\[
  f(X) \mathrel{\mathring{\ll}} Y' \leq f(Y) \quad \text{dans } \mathcal{M}'
\]

\[
  X \mathrel{\mathring{\ll}} Y_1 \leq Y
\]

\[
  (**') \qquad f(X) \ll f(Y) \Longrightarrow X \ll Y
\]

\[
  f(X) \mathrel{\mathring{\ll}} f(Y_1) \leq f(Y) \quad \text{donc } Y' = f(Y_1), \text{ OK.}
\]

\[
  f : \mathcal{M} \longrightarrow \text{Figél}(P),
\]

\[
  \mathcal{M} \longrightarrow \mathcal{P}(P), \qquad X \longmapsto |X| \overset{\mathrm{def}}{=} |f(X)|,
\]

\[
  (\alpha) \qquad f(X) = \{ |Y| \mid Y \in \widetilde{X} \}.
\]

\[
  X \longmapsto |X| : \mathcal{M} \longrightarrow \mathcal{P}(P),
\]

\[
  |\Phi| = \bigcup_{X \in \Phi} |X|,
\]

\[
  |\partial X| \subset |X|, \qquad |\partial X| = \bigcup_{Y < X} |Y|
\]

\[
  \partial X = \widetilde{X} - \{X\} = \{ Y \in \mathcal{M} \mid Y < X \} \ ),
\]

\[
  |X|^{\circ} = |X| \setminus |\partial X|.
\]

\[
  |X|^{\circ} = f(X)^{\circ},
\]

\[
  X \mathrel{|\circ|} Y \Longrightarrow f(X) \mathrel{|\circ|} f(Y).
\]

\[
  X \mathrel{\mathring{\ll}} Y \Longrightarrow f(X) \mathrel{\mathring{\ll}} f(Y),
\]

\[
  \varphi : \widetilde{X} \longrightarrow \widetilde{Y}, \qquad \varphi(X) = Y,
\]

\[
  X' \mathrel{\mathring{\ll}} \varphi(Y')
\]

\[
  X \mathrel{\mathring{\ll}} Y \Longrightarrow |X|^{\circ} \subset |Y|^{\circ}
\]

\[
  |X|^{\circ} \cap |\partial Y| = \emptyset
\]

\[
  |X|^{\circ} \cap |Z|^{\circ} = \emptyset \qquad \forall Z \not\leq Y
\]

\[
  |Z| = \bigcup_{Z' \leq Z} |Z'|^{\circ}
\]

\[
  |X|^{\circ} \cap |Z|^{\circ} = \emptyset \qquad \forall Z < Y
\]

\[
  X \mathrel{|\circ|} Z \qquad \text{(via (ii))}
\]

\[
  Z < Y \text{ \ill{} } Z \mathrel{|\circ|} Y \quad (\mathrm{M}_0 4)
\]

\[
  X \mathrel{|\circ|} Z \quad (\mathrm{M}_0 3)\text{], on a gagné !}
\]

\[
  \mathcal{M} \longrightarrow \operatorname{Figures}(P)^{*}
\]

\[
  \mathcal{M} \longrightarrow \mathcal{P}(P), \qquad X \longmapsto |X|
\]

\[
  (\text{ii fid}) \qquad \forall X, Y \in \mathcal{M}, \quad X \mathrel{|\circ|} Y \Longleftrightarrow |X|^{\circ} \cap |Y|^{\circ} = \emptyset
\]

\[
  X \longmapsto \mathrm{Omb}(X) \qquad \mathcal{M} \xrightarrow[\text{fidèle}]{\ \ } {}^{*}\mathcal{P}(\mathcal{M})
\]

\[
  X \mathrel{|\circ|} Y \text{ ssi } \nexists Z,\ Z \mathrel{\mathring{\ll}} X,\ Z \mathrel{\mathring{\ll}} Y,
\]

\[
  |X| \overset{\mathrm{def}}{=} \widetilde{X} = \mathcal{M}_{\leq X} = \mathrm{Omb}(X)
\]

\[
  |X|^{\circ} = \{ X \} \qquad \text{pour } \forall X \in \mathcal{M}
\]

\[
  \operatorname{cosupp}^{\circ}(A) = \{ \text{\struck{$X$}}\, Y \in \mathcal{M} \mid A \mathrel{|\circ|} \text{\struck{$X$}}\, Y \}
\]

\[
  A \mathrel{|\circ|} B \overset{\mathrm{def}}{\Longleftrightarrow} \forall X \in A,\ Y \in B, \text{ on a } X \mathrel{|\circ|} Y
\]

\[
  A \mathrel{|\circ|} X \overset{\mathrm{def}}{\Longleftrightarrow} A \mathrel{|\circ|} \{ X \}.
\]

\[
  \begin{cases}
    \operatorname{cosupp}^{\circ}\bigl(\bigcup A_i\bigr) = \bigcap_i \operatorname{cosupp}^{\circ}(A_i) \\
    \operatorname{cosupp}^{\circ}(\emptyset_{\mathcal{M}}) = \mathcal{M} \\
    \operatorname{cosupp}^{\circ}(\mathcal{M}) = \emptyset
  \end{cases}
\]

\[
  \operatorname{supp}^{\circ}(A) = \operatorname{cosupp}^{\circ}\bigl(\operatorname{cosupp}^{\circ}(A)\bigr)
\]

\[
  A = \operatorname{supp}^{\circ}(A)
\]

\[
  X \in A, \quad X' \mathrel{\mathring{\ll}} X \Longrightarrow X' \in A.
\]

\[
  \operatorname{cosupp} X = \{ Y \in \mathcal{M} \mid \widetilde{Y} \subset \operatorname{cosupp}^{\circ}(\widetilde{X}) \}
\]

\[
  \operatorname{cosupp} A = \{ Y \in \mathcal{M} \mid \widetilde{Y} \subset \operatorname{cosupp}^{\circ}(\widetilde{A}) \}
\]

\[
  \Sigma_{\mathcal{M}} \xrightarrow{\ \sim\ } \Sigma_{\mathcal{M}}, \qquad S \longmapsto \text{\struck{$\mathcal{C}$}}\, \mathrm{c}S,
\]

\[
  \operatorname{Sup} S_i = \operatorname{supp}^{\circ}\Bigl(\bigcup S_i\Bigr)
\]

\[
  \Sigma_{\mathcal{M}} \longrightarrow \Sigma_{\mathcal{L}}, \qquad S \longmapsto S \cap \mathcal{L}
\]

\[
  \operatorname{cosupp}^{\circ}_{\mathcal{M}} X = \operatorname{cosupp}^{\circ}_{\mathcal{M}}(\mathrm{Omb}^{\circ} X)
\]

\[
  \operatorname{cosupp}^{\circ}_{\mathcal{M}}(A) = \operatorname{cosupp}^{\circ}_{\mathcal{M}}(\mathrm{Omb}^{\circ}(A))
\]

\[
  \mathrm{Omb}^{\circ}(A) = \bigcup_{X \in A} \mathrm{Omb}^{\circ}(X) \subset \mathcal{L}
\]

\[
  \operatorname{cosupp}^{\circ}_{\mathcal{M}}(A) \cap \mathcal{L} = \operatorname{cosupp}^{\circ}_{\mathcal{L}}(\mathrm{Omb}(A)),
\]

\[
  \Sigma_{\mathcal{L}} \longrightarrow \Sigma_{\mathcal{M}}
\]

\[
  A \longmapsto \operatorname{supp}^{\circ}_{\mathcal{M}}(A) \overset{?}{=} \{ X \in \mathcal{M} \mid \mathrm{Omb}^{\circ} X \subset A \}
\]

\[
  \mathrm{Omb}^{\circ}(X) \subset A \Longrightarrow \operatorname{supp}^{\circ}(X) \subset \operatorname{supp}^{\circ}_{\mathcal{M}}(A) \quad \text{d'où}
\]

\[
  X \mathrel{|\circ|} Y \Longleftrightarrow o(X) \parallel o(Y)
\]

\[
  \varphi(A) = \{ X \in \mathcal{M} \mid o(X) \subset A \}
\]

\[
  \Sigma_{\mathcal{L}} \rightleftarrows \Sigma_{\mathcal{M}} \quad \dots
\]

\[
  \Sigma_{\mathcal{M}} \simeq \mathcal{P}(\mathcal{L})
\]

\[
  \mathcal{M} \longrightarrow \mathcal{P}(P), \qquad X \longmapsto |X|
\]

\[
  \Sigma_{\mathcal{M}} \hookrightarrow \mathcal{P}(P)
\]

\[
  \varphi : X \longmapsto |X|^{\circ} \qquad \mathcal{M} \longrightarrow \mathcal{P}(P),
\]

\[
  |A|^{\circ} \text{ ou } \varphi(A) \overset{\mathrm{def}}{=} \bigcup_{X \in A} \underbrace{\varphi(X)}_{|X|^{\circ}}
\]

\[
  A \parallel B \Longleftrightarrow \underbrace{\varphi(A) \cap \varphi(B)}_{S} = \emptyset
\]

\[
  \operatorname{cosupp}^{\circ}(A) = \{ \text{\struck{$X$}}\, X \in \mathcal{M} \mid \text{\struck{$\varphi(\ldots)$}}\ X \subset P \setminus \varphi(A) \}
\]

\[
  \varphi(\underbrace{\operatorname{cosupp}^{\circ}(A)}_{S}) \subset P \setminus \varphi(A)
\]

\[
  X \in S \Longleftrightarrow \varphi(X) \subset \varphi(S)
\]

\[
  \Sigma_{\mathcal{M}} \longrightarrow \mathcal{P}(P), \qquad S \longmapsto \varphi(S) \ (= |S|^{\circ})
\]

\[
  \text{\struck{$S_Q$}} = \{ X \in \mathcal{M} \mid \varphi(X) \subset Q \}
\]

\[
  S_Q = \operatorname{supp}^{\circ}\bigl(\{ X \in \mathcal{M} \mid \varphi(X) \subset Q \}\bigr).
\]

\[
\text{\struck{$X,Y\in F \Rightarrow X \not\lessgtr Y$}}
\]

\[
\mathrm{supp}^\circ X = A \subset \mathcal{M} \qquad (X\in\mathcal{M})
\]

\[
|X|^\circ \subset P .
\]

\[
|X|^\circ \subset |\mathrm{supp}^\circ X|^\circ .
\]

\[
|X|^\circ = |\mathrm{supp}^\circ X|^\circ
\]

\[
\forall Y \in \mathrm{supp}^\circ X, \text{ on ait } |Y|^\circ \subset |X|^\circ
\]

\[
\begin{gathered}
|Y|^\circ \not\subset |X|^\circ \Longrightarrow Y \notin \mathrm{supp}^\circ X, \text{ i.e.}\\
\exists Z\in\mathcal{M} \text{ avec } Z \mathrel{|{\circ}|} X,\; Z \mathrel{\overline{|\circ|}} Y,
\end{gathered}
\]

\[
\begin{cases}
|Z|^\circ \cap |X|^\circ = \emptyset\\
|Z|^\circ \cap |Y|^\circ \neq \emptyset .
\end{cases}
\]

\[
|Y|^\circ \cap \partial X \neq \emptyset ,
\]

\[
\partial X = \bigcup_{Z<X} |Z|^\circ ,
\]

\[
\exists Z < X, \quad |Y|^\circ \cap |Z|^\circ \neq \emptyset ,
\]

\[
|Y|^\circ \cap \complement |X| \neq \emptyset
\]

\[
\exists Z \in \mathcal{M} \text{ \struck{$\complement|X|$} avec } x\in|Z|, \quad Z \parallel X \quad (\text{i.e. } |Z|\cap|X| = \emptyset) .
\]

\[
P \setminus |X| = \bigcup_{Z\in\mathrm{cosupp}^\circ(X)} |Z| \qquad \text{ou}
\]

\[
\Bigl(= \bigcup_{Z\in\mathrm{cosupp}^\circ(X)} |Z|^\circ\Bigr)
\]

\[
P \setminus |X|^\circ = \bigcup_{Z\in\mathrm{cosupp}^\circ X} |Z|^\circ
\]

\[
\begin{aligned}
\varphi(\mathrm{supp}^\circ X) &\overset{\text{déf}}{=} |\mathrm{supp}^\circ X|^\circ = |X|^\circ\\
\varphi(\mathrm{cosupp}^\circ X) &\overset{\text{déf}}{=} |\mathrm{cosupp}^\circ X|^\circ = P\setminus |X|^\circ .
\end{aligned}
\]

\[
\begin{aligned}
\varphi(\mathrm{supp}\,X) &= |X|\\
\varphi(\mathrm{cosupp}\,X) &= P\setminus |X|
\end{aligned}
\]

\[
\begin{aligned}
\mathrm{supp}\,X &= \mathrm{supp}^\circ(\widetilde{X})\\
\mathrm{cosupp}\,X &= \mathrm{cosupp}^\circ(\widetilde{X})
\end{aligned}
\]

\[
|X| \subset \varphi(\mathrm{supp}\,X) .
\]

\[
Y \in \mathrm{supp}\,X \;(= \mathrm{supp}^\circ(\widetilde{X})) \Longrightarrow |Y|^\circ \subset |X|
\]

\[
|Y|^\circ \not\subset |\overline{X}| \Longrightarrow Y \notin \mathrm{supp}^\circ(\widetilde{X}) \quad \text{i.e. } \exists Z
\]

\[
\text{i.e. } |Y|^\circ \cap (P \setminus \overline{X}) \neq \emptyset
\qquad
\text{avec } X \mathrel{|{\circ}|} Z,\; Y \mathrel{\overline{|\circ|}} Z ,
\]

\[
\text{i.e. } |X| \cap |Z|^\circ \neq \emptyset,\; |Y|^\circ \cap |Z|^\circ .
\]

\[
\mathrm{supp}^\circ(\{Y,X\}) \quad\text{et}\quad \varphi(\mathrm{supp}^\circ(\{Y,X\}))
\]

\[
\text{\struck{\ill{}}}\; |X|^\circ \cup |Y|^\circ \subset \varphi(\mathrm{supp}^\circ(\{Y,X\}))
\]

\[
A = \text{\struck{\ill{}}}\; \mathrm{supp}^\circ(\Phi) .
\]

\[
X \longmapsto |X|, \quad \mathcal{M} \longrightarrow \mathfrak{P}(P)
\]

\[
\forall F \in \mathfrak{F}, \text{ on a } \varphi(\mathrm{cosupp}^\circ(F)) = X \setminus |F|
\]

\[
X\setminus|F| = \bigcup_{\substack{Z\in\mathcal{M}\\ \text{t.q. } Z \mathrel{|{\circ}|} F}} |Z|^\circ
\qquad [\text{i.e. } |Z|^\circ \subset X\setminus|F|]
\]

\[
\boxed{\forall x \in X - |F|, \;\exists Z \text{ avec } x \in |Z|^\circ \subset X\setminus|F|}
\]

\[
\varphi(\mathrm{supp}^\circ\Phi) = |\Phi|^\circ \Bigl(\overset{\text{déf}}{=} \bigcup_{X\in\Phi} |X|^\circ\Bigr) \qquad (*)
\]

\[
\varphi(\Phi) \overset{\text{déf}}{=} |\Phi|^\circ \subset \varphi(\mathrm{supp}^\circ(\Phi)) ,
\]

\[
\text{\uncertain{Soit}}\quad Y \in \mathrm{supp}^\circ(\Phi) \Longrightarrow |Y|^\circ \subset |\Phi|^\circ \quad \text{i.e.}
\]

\[
\begin{gathered}
(*)\quad |Y|^\circ \not\subset |\Phi|^\circ \Longrightarrow Y \notin \mathrm{supp}^\circ(\Phi), \text{ i.e. } \exists Z \ldots\\
Z \mathrel{|{\circ}|} \Phi, \text{ et } Z \mathrel{\overline{|\circ|}} Y, \text{ i.e.}\\
|Z|^\circ \cap |\Phi|^\circ = \emptyset,\; |Z|^\circ\cap|Y|^\circ \neq \emptyset .
\end{gathered}
\]

\[
|F|\setminus|\Phi|^\circ = |F\setminus\Phi|^\circ ,
\]

\[
\Phi \longmapsto \mathrm{supp}^\circ(\Phi) : \mathfrak{P}(F) \longrightarrow \Sigma_{\mathcal{M}}
\]

\[
\begin{aligned}
\mathrm{supp}^\circ\Bigl(\bigcap_i \Phi_i\Bigr) &= \bigcap_i \mathrm{supp}^\circ(\Phi_i) && I \neq \emptyset\\
\mathrm{supp}^\circ\Bigl(\bigcup_i \Phi_i\Bigr) &= \operatorname{Sup}_i(\mathrm{supp}^\circ(\Phi_i)) && \text{sup dans } \Sigma_{\mathcal{M}}
\end{aligned}
\]

\[
\mathrm{supp}^\circ(\Psi\setminus\Phi) = \mathrm{supp}^\circ\Psi \cap \complement\,\mathrm{supp}^\circ\Phi
\]

\[
\begin{gathered}
A \longmapsto |A|^\circ = \bigcup_{X\in A} |X|^\circ\\
\Sigma_{\mathcal{M},F} \longrightarrow \mathfrak{P}(|F|) \subset \mathfrak{P}(P)
\end{gathered}
\]

\[
\begin{aligned}
A \cap \operatorname{Sup}_i B_i &= \operatorname{Sup}_i (A\cap B_i)\\
A \setminus (A\setminus B) &= \text{\struck{$A\cap$}}\, B \qquad (\text{si } B\subset A)
\end{aligned}
\]

\[
A\cap B = \emptyset \Longleftrightarrow A \mathrel{|{\circ}|} B ;
\]

\[
\mathrm{supp}^\circ(F) \cap \mathrm{cosupp}^\circ\Phi = \mathrm{supp}^\circ(\Phi')
\]

\[
\text{\struck{si}}\; Z \in \mathrm{supp}^\circ(F),\; \text{\struck{et}}\; Z \parallel \Phi \Longrightarrow Z \in \mathrm{supp}^\circ(\Phi')
\]

\[
\forall a,b\in L, \quad a<b \Longrightarrow \;]a,b[\; \neq \emptyset
\qquad \Bigl(]a,b[ \;\overset{\text{déf}}{=} \{x\in L \mid a<x<b\}\Bigr)
\]

\[
\mathcal{M} \subset \text{\struck{$\mathfrak{P}(\mathfrak{P}(L))$}}\; \mathrm{Fig}(L)
\]

\[
\begin{cases}
\mathcal{M} = \mathcal{M}_0 \sqcup \mathcal{M}_1, \quad \text{avec}\\
\mathcal{M}_0 = \{P_x \mid x \in L\}, \quad \text{où } P_x = \{\{x\}\}\\
\mathcal{M}_1 = \{S_{a,b} \mid a,b\in L,\; a<b\}, \quad S_{ab} = \{[a,b],\{a\},\{b\}\}
\end{cases}
\]

\[
\begin{cases}
\partial P_x = \emptyset\\
\partial S_{a,b} = \{P_a, P_b\}
\end{cases}
\]

\[
\begin{cases}
\text{a) si } X = P_x,\, Y = P_y : & x = y \quad \text{i.e. } P_x = P_y\\
\text{b) si } X = P_x,\, Y = S_{ab} : & a<x<b \quad [\text{i.e. } x\in|X|^\circ]\\
\text{c) si } X = S_{a,b},\, Y = S_{a',b'} : & a'\le a<b\le b'
\end{cases}
\]

\[
[\text{i.e. } |X|\subset|Y|,\; |X|^\circ\subset|Y|^\circ]
\]

\[
X \mathrel{|{\circ}|} Y \Longleftrightarrow
\begin{cases}
X = P_x,\, Y = P_y : & [x\neq y \text{ i.e.}]\; x<y \text{ ou } y<x\\
X = P_x,\, Y = S_{a,b}\;(\text{ou l'inverse}) : & x\le a \text{ ou } b \le x\\
X = S_{a,b},\, Y = S_{a',b'} : & a<b\le a'<b' \text{ ou } a'<b'\le a<b
\end{cases}
\]

\[
\begin{cases}
\mathcal{M} \subset \mathfrak{P}(\mathfrak{P}(L))\\
\mathcal{M} = \mathcal{M}_0 \sqcup \mathcal{M}_1\\
\mathcal{M}_0 = \{P_x \mid x\in L\}, \quad P_x = \{\{x\}\}\\
\mathcal{M}_1 = \{S_{a,b} \mid a,b\in L,\; a<b\}, \quad S_{a,b} = \{[a,b],\{a\},\{b\}\}
\end{cases}
\]

\[
|S_{a,b}|^\circ = [a,b]\setminus\{a,b\} = \;]a,b[
\]

\[
S_{a,b} \overset{\circ}{\ll} S_{a',b'} \Longleftrightarrow a'\le a<b\le b' \Longleftrightarrow |S_{a,b}|^* \subset |S_{a',b'}|^*
\]

\[
\Longrightarrow |S_{a,b}|^\circ \subset |S_{a',b'}|^\circ
\]

\[
P_x \mathrel{|{\circ}|} P_y \Longleftrightarrow x<y \text{ ou } y<x \text{ i.e. } \{x,y\}\in\mathrm{Drap}_2(L) \Longrightarrow x\neq y
\]

\[
P_x \mathrel{|{\circ}|} S_{a,b} \Longleftrightarrow x\le a \text{ ou } y\ge b \Longrightarrow |P_x|^\circ\cap|S_{a,b}|^\circ = \emptyset
\]

\[
S_{a,b} \mathrel{|{\circ}|} S_{a',b'} \Longleftrightarrow a<b\le a'<b' \text{ ou } a'<b'\le a<b
\]

\[
\Longrightarrow |S_{a,b}|^\circ \cap |S_{a',b'}|^\circ = \emptyset .
\]

\[
X \ll Y \Longleftrightarrow
\begin{cases}
X = P_x,\, Y = P_y, & x = y\\
X = P_x,\, Y = S_{a,b}, & a\le x\le b \quad \text{\struck{$x\in[a,b]$ ($=|S_{a,b}|$)}}\\
X = S_{a,b},\, Y = S_{a',b'} : & a'\le a<b\le b' ,
\end{cases}
\]

\[
\begin{cases}
X \ll Y \Longleftrightarrow |X|\subset|Y|\\
\text{si } X,Y\in\mathcal{M}_1, \quad X\ll Y \Longleftrightarrow X \overset{\circ}{\ll} Y
\end{cases}
\]

\[
\begin{aligned}
&\text{Mag 3 :} && X \mathrel{|{\circ}|} Y,\; X'\overset{\circ}{\ll}X,\; Y'\overset{\circ}{\ll}Y \Longrightarrow X'\mathrel{|{\circ}|}Y' && \text{OK}
\end{aligned}
\]

\[
\text{Mag L 1} \Longleftrightarrow \forall S_{a,b},\; |S_{a,b}|^\circ \overset{\text{déf}}{=} \;]a,b[\; \neq \emptyset
\]

\[
\begin{aligned}
&\text{a) pour } P_x, S_{a,b} : && x \parallel \;]a,b[\; \Longrightarrow x\le a \text{ ou } x\ge b\\
&\text{b) pour } S_{a,b}, S_{a',b'} : && ]a,b[\; \parallel \;]a',b'[\; \Longrightarrow b\le a' \text{ ou } b'\le a
\end{aligned}
\]

\[
x \parallel \;]a,b[\; \Longrightarrow \text{\struck{$x\parallel\{a,b\}$}}\; x\le a \text{ ou } x\ge b .
\]

\[
\text{\struck{$x\parallel A \Longleftrightarrow \forall y\in A,\; x\parallel y$ (i.e. $x<y$ ou $y<x$))}}
\]

\[
L_{\le a} = \text{\struck{$\bigcap_{b\in L_{>a}}$}}
\]

\[
\bigcap_{c>a} L_{\le c} = L_{\le a}
\]

\[
\bigcap_{c<b} L_{\ge c} = L_{\ge b}
\]

\[
(*) \qquad \mathrm{Omb}^\circ(X) \cap \mathrm{Omb}^\circ(Y) = \emptyset \Longrightarrow X \mathrel{|{\circ}|} Y \quad ?
\]

\[
P_x \neq P_y \Longrightarrow x \mathrel{|{\circ}|} y
\]

\[
\begin{aligned}
&a'\le a<b\le b' && \text{on prend } S_{ab} \text{ comme \uncertain{\ill{}} intérieur commun}\\
&a\le a'<b\le b' \text{ ou } a'\le a<b'\le b && \text{on prend } S_{a'b} \text{ resp. } S_{ab'}\\
&a\le a'<b'\le b && \text{on prend } S_{a',b'}
\end{aligned}
\]

\[
P_x \not\lessgtr Y \Longleftrightarrow P_x = Y \text{ ou } P_x \mathrel{|{\circ}|} Y
\]

\[
\begin{aligned}
P_x \not\lessgtr P_y &\Longleftrightarrow \{x,y\}\in\mathrm{Drap}(L)\\
P_x \not\lessgtr S_{a,b} &\Longleftrightarrow P_x \mathrel{|{\circ}|} S_{a,b} \Longleftrightarrow x\le a \text{ ou } x\ge b
\end{aligned}
\]

\[
S_{a,b}\not\lessgtr S_{a',b'} \Longleftrightarrow
\begin{cases}
\text{ou (i) } S_{a,b} = S_{a',b'} \quad \text{i.e. } a = a',\, b = b'\\
\text{ou (ii) } S_{ab} \mathrel{|{\circ}|} S_{a',b'}\\
\text{ou (iii) } a<b = a'<b' \text{ ou } a'<b' = a<b
\end{cases}
\]

\[
S_{a,b}\not\lessgtr S_{a',b'} \Longleftrightarrow S_{a,b} = S_{a',b'} \text{ ou } S_{a,b} \mathrel{|{\circ}|} S_{a',b'}
\]

\[
a<b\le a'<b' \text{ ou } a'<b'\le a<b
\]

\[
\delta X =
\begin{cases}
\mathbf{x} & \text{si } X = P_x\\
|\partial X| = \{a,b\} & \text{si } X = S_{a,b}
\end{cases}
\]

\[
\delta X \subset L, \quad \operatorname{card}\delta X \in \{1,2\}
\]

\[
X\not\lessgtr Y \Longrightarrow \delta(X)\cup\delta(Y) \in \mathrm{Drap}(L)
\]

\[
\begin{cases}
\text{lieux égaux ou \struck{\ill{}} disjoints}\\
\text{lieu \struck{\ill{}} incident à un segment, ou disjoint des segments}\\
\text{segments disjoints, segments \textit{adjacents}}
\end{cases}
\]

\[
T = \{t_1,t_2,\ldots,t_n\} \quad \text{avec } t_1<t_2<\ldots<t_n ,
\]

\[
\delta F = \bigcup_{X\in F} \delta X .
\]

\[
F_T = \text{ensemble des toutes les structures interstitielles de } T .
\]

\[
\text{\struck{$F = \widetilde{S}_{a,b} = \{S_{a,b}, P_a, P_b\}$ .}}
\]

\[
\text{\struck{$\mathfrak{S}_a = \mathrm{supp}^\circ P_a$, $\mathfrak{S}_b = \mathrm{supp}^\circ P_b$,}}
\]

\[
F_T = \{X\in\mathcal{M} \mid X \text{ interstitiel par } T\} ,
\]

\[
\begin{aligned}
T = \emptyset &\Longrightarrow F_T = \emptyset && \text{graphisme : ?}\\
T = \{a\} &\Longrightarrow F_T = \{P_a\} && \text{graphisme : un point } a\\
T = \{a,b\} &\Longrightarrow F_T = \widetilde{S}_{a,b} && \text{graphisme : un segment } a \text{---} b
\end{aligned}
\]

\[
T < T' \quad \text{ssi} \quad \forall t\in T,\; t'\in T', \text{ on a } t<t' ,
\]

\[
F_T \parallel F_{T'} \quad (\Longleftrightarrow F_T \mathrel{|{\circ}|} F_{T'})
\]

\[
\underset{\text{dans } \mathrm{Drap}^*(L)}{T\parallel T'} \quad \text{i.e. } \{T,T'\}\in\mathrm{Drap}^*(\mathrm{Drap}^*(L)) .
\]

\[
F_{\mathcal{T}} \overset{\text{déf}}{=} \bigcup_{T\in\mathcal{T}} F_T
\]

\[
F_T = F_{\{T\}}
\]

\[
\begin{gathered}
\mathcal{T} \longmapsto F_{\mathcal{T}}\\
\mathrm{Drap}(\mathrm{Drap}^*(L)) \longrightarrow \mathrm{Fig\,fin}(\mathcal{M}) = \mathfrak{F}
\end{gathered}
\]

\[
\text{\struck{$\mathrm{Supp}^\circ(\Phi\cup\Psi) = \mathrm{Supp}^\circ(\Phi)$}}
\]

\[
\mathrm{supp}(X) \Bigl(\overset{\text{déf}}{=} \mathrm{supp}^\circ(\widetilde{X})\Bigr) = \mathrm{Omb}(X) = \{Z\in\mathcal{M} \mid |Z|\subset|X|\}
\]

\[
\mathrm{Omb}(X) \subset \mathrm{supp}(X)
\]

\[
\mathrm{supp}(\{P_a\}) = \{P_a\}
\]

\[
\mathrm{supp}(\{S_{a,b}\}) = \Bigl\{ Z \Bigm|
\begin{array}{l}
Z = P_x \text{ avec } a\le x\le b\\
\text{ou } Z = S_{x,y} \text{ avec } a\le x<y\le b
\end{array}
\Bigr\}
\]

\[
\mathrm{supp}^\circ\{S_{ab}\} = \mathrm{supp}(\{S_{ab}\}) \setminus \partial S_{ab} \qquad (\partial S_{ab} = \{P_a, P_b\})
\]

\[
= \Bigl\{ Z \Bigm|
\begin{array}{l}
Z = P_x \text{ avec } a<x<b\\
\text{ou } Z = S_{x,y} \text{ avec } a\le x<y\le b
\end{array}
\Bigr\}
\;\; \text{\struck{$= \mathrm{Omb}^\circ(S_{ab})$}}
\]

\[
\begin{aligned}
\mathrm{supp}^\circ(\{S_{a,b}, P_a\}) &= \mathrm{Omb}^\circ(S_{ab}) \cup \{P_a\}\\
\mathrm{supp}^\circ(\{S_{ab}, P_b\}) &= \mathrm{Omb}^\circ(S_{ab}) \cup \{P_b\}
\end{aligned}
\]

\[
\text{\struck{$\mathrm{cosupp}^\circ(P_a) = L_{<a}\cup L_{>a} = \{x\in L \mid x<a \text{ ou } x>a\}$}}
\]

\[
\text{\struck{$\mathrm{supp}^\circ(P_a) = \mathrm{supp}(P_a) = \mathrm{cosupp}^\circ(\mathrm{cosupp}^\circ(P_a))$}}
\]

\[
\text{\struck{$X \mathrel{|{\circ}|} x \quad \forall x\in\mathrm{cosupp}^\circ(P_a)$}}
\]

\[
X \mathrel{|{\circ}|} Y, \quad \forall Y\in\mathrm{cosupp}^\circ(P_a) .
\]

\[
\mathrm{cosupp}^\circ(P_a) = \mathcal{M} - \{P_a\} \quad (\text{car tout } X\in\mathcal{M}
\]

\[ x<a<y . \]

\[ \delta X\geq a, \quad\text{puis}\quad \delta X\leq b \]

\[ x<a<b \]

\[ = \operatorname{cosupp}^\circ(P_b) = \mathcal{M}\setminus\{P_b\} \]

\[ \operatorname{cosupp}^\circ(\{P_a\}) = \{P_b\text{\struck{$\}$}},\ S_{a,b},\ S_{c,b} \]

\[ \operatorname{supp}^\circ(\{P_a\}) = \{P_a,\ P_b\} \neq \{P_a\} \]

\[ \operatorname{cosupp}^\circ(\widetilde{S}_{a,b}) = \emptyset \]

\[ \operatorname{supp}^\circ(\widetilde{S}_{a,b}) = \mathcal{M} \neq \operatorname{Omb}(S_{a,b}) \]

\[ \text{\struck{$\operatorname{supp}^\circ(\Phi\cup\Psi)=\operatorname{supp}^\circ\Phi\cup\operatorname{supp}^\circ\Psi$.}} \]

\[ \partial F_T \overset{\text{déf}}{=} \partial T = \{\text{ensemble formé du plus petit et du plus grand élément de } T=F_0\} \]

\[ \Phi = F\setminus\partial F \]

\[ \Phi = F\setminus\alpha \]

\[ \text{Préfigcomm}(\mathcal{M}) \]

\[ \Phi < \Phi' \iff
\begin{cases}
\text{pour tout } X\in\Phi,\ X'\in\Phi',\ \text{on a } X \underset{\mathrm{pos}}{<} X' \\
\text{a) [\dots]}\ \operatorname{card}(\delta\Phi\cap\delta\Phi')\leq 1 \\
\text{b) si } x\in\partial\Phi\cap\partial\Phi',\ \text{on a } x\notin\Phi,\ x\notin\Phi'
\end{cases} \]

\[ \delta\Phi \overset{\text{déf}}{=} \delta F = F\cap\mathcal{M}_0
= \{a\in\mathcal{L} \mid P_a\in F,\ \text{i.e. } \exists X\in\Phi,\ P_a\leq X\} \]

\[ \text{\struck{$\partial\Phi \overset{\text{déf}}{=} \partial F =$}}\quad
\text{\struck{$\emptyset$ si $\operatorname{card}\delta F\leq 1$ ; ensemble formé des plus petit et plus grand}} \]

\[ \partial\Phi \overset{\text{déf}}{=}
\begin{cases}
\text{a) } \emptyset \text{ si } \operatorname{card}\delta\Phi\leq 1, \text{ i.e. } \Phi=\emptyset \text{ ou } \{P_a\} \\
\text{b) bord de } \delta\Phi \text{ dès } \operatorname{card}\delta\Phi\geq 2
\end{cases} \]

\[ \Phi^{\bullet} =
\begin{cases}
\partial\Phi & \text{si } \operatorname{card}\delta\Phi\geq 2 \\
\{a\} & \text{si } \Phi=\{P_a\}, \text{ i.e. } \operatorname{card}\delta\Phi=1 \\
\emptyset & \text{si } \Phi=\emptyset, \text{ i.e. } \operatorname{card}\delta\Phi=0
\end{cases} \]

\[ \partial\Phi = \bigcup \partial\Phi_i \]

\[ \Phi^{\bullet} = \bigcup \Phi_i^{\bullet} \]

\[ \Phi = \{S_{a,b},\ S_{b,c},\ P_d\} \]

\[ \delta\Phi = \{a,b,c,d\} \]

\[ \partial\Phi = \{a,b,c\} \]

\[ \Phi^{\bullet} = \{a,b,c,d\} \]

\[ \operatorname{ordre}(a,\Psi) = \operatorname{card}(\{X\in\mathcal{M}_1\cap\Psi \mid a\triangleleft X\}) \]

\[ \partial\Phi = \{a\in\mathcal{L} \mid \operatorname{ordre}(a,\overline{\Phi})=1\} \]

\[ \Phi = \{S_{a,b},\ P_b,\ S_{b,c},\ P_d\}, \]

\[ \delta\Phi = \{a,b,c,d\} \]

\[ \partial\Phi = \{a,c\} \]

\[ \Phi^{\bullet} = \{a,c,d\} \]

\[ \partial\Phi \subset \Phi^{\bullet} \subset \delta\Phi \]

\[ \partial\Phi = \{a\in\delta\Phi \mid \exists X\in\Phi \text{ tel que } P_a<X, \text{ et de plus, quand } P_a\in\Phi, \text{ cet } X \text{ est unique}\} \]

\[ = \text{ens. des sommets non isolés de } \Phi, \]

\[ \Phi^{\bullet} = \partial\Phi \amalg \Phi_{\mathrm{isol}} \]

\[ X \underset{\mathrm{pos}}{<} Y \iff
\begin{cases}
\text{a) } X=P_x,\ Y=P_y & x<y \\
\text{b) } X=P_x,\ Y=S_{a,b} & x\leq a<b \\
\text{c) } X=S_{a,b},\ Y=P_y & a<b\leq y \\
\text{d) } X=S_{a,b},\ Y=S_{a',b'} & a<b\leq a'<b'
\end{cases} \]

\[ X \underset{\mathrm{pos}}{<} Y \iff X\,|{\circ}|\,Y \text{ et } \operatorname{or}(\delta X) < \operatorname{ex}(\delta Y) \]

\[ X\,|{\circ}|\,Y \iff X \underset{\mathrm{pos}}{<} Y \ \text{ou}\ Y \underset{\mathrm{pos}}{<} X \]

\[ \Phi \underset{\mathrm{pos}}{<} \Psi \overset{\text{déf}}{\iff} \forall X\in\Phi,\ Y\in\Psi,\quad X<Y . \]

\[ \Phi \underset{\mathrm{pos}}{\overset{1}{<}} \Psi \overset{\text{déf}}{\iff}
\begin{cases}
\Phi \underset{\mathrm{pos}}{<} \Psi, \text{ et dans le cas où} \\
\operatorname{ex}\Phi = \operatorname{or}\Psi \ (\text{\uncertain{noté} } a), \text{ on a} \\
P_a\notin\Phi,\ P_a\notin\Psi
\end{cases} \]

\[ \Phi \underset{\mathrm{pos}}{\leq} \Psi \iff \operatorname{ex}\Phi < \operatorname{or}\Psi, \]

\[ D = \{\Phi_1 \underset{\mathrm{pos}}{\overset{1}{<}} \Phi_2 \underset{\mathrm{pos}}{\overset{1}{<}} \cdots \underset{\mathrm{pos}}{\overset{1}{<}} \Phi_n\} \]

\[ D \in \operatorname{Drap}^*_{\overset{1}{<}}(\text{Préfigcomm}(\mathcal{M})) \]

\[ \Phi_D = \bigcup_{\Phi\in D}\Phi \ \text{\struck{$\ill{}$}}\ = \bigcup_{i\in[1,n]}\Phi_i \]

\[ \Phi \underset{\mathrm{pos}}{\leq} \Psi . \]

\[ \text{que}\quad \Phi \overset{1}{<} \Psi \ \text{ou}\ \Psi \overset{1}{<} \Phi, \ \text{i.e.} \]

\[ \{\Phi,\Psi\} \in \operatorname{Drap}_{\overset{1}{<}}(\text{Préfigcomm}(\mathcal{M})) . \]

\[ \Phi_{D,\alpha} = F_D\setminus\alpha , \]

\[ (D,\alpha) \longmapsto \Phi_{D,\alpha} \]

\[ \text{\struck{$\alpha=\partial D$}}\quad \operatorname{card}D\geq 2 \ \text{et}\ \alpha = \partial D - (\partial D\cap\Theta) . \]

\[ D = \{\Phi_1 \overset{1}{<} \Phi_2 \overset{1}{<} \cdots \overset{1}{<} \Phi_n\} \]

\[ D \in \operatorname{Drap}^*_{\overset{1}{<}}(\text{Préfigcomm}(\mathcal{M})), \]

\[ \Phi_D = \bigcup_{\Phi\in D}\Phi = \bigcup_{1\leq i\leq n}\Phi_i . \]

\[ D \longmapsto \Phi_D = \bigcup_{\Phi\in D}\Phi \]

\[ \operatorname{Drap}^*_{\overset{1}{<}}(\text{Préfigcomm}(\mathcal{M})) \longrightarrow \text{Préfig}(\mathcal{M}) \]

\[ \operatorname{supp}^\circ\Phi = \bigcup_i \operatorname{supp}^\circ(\Phi_i) \]

\[ \operatorname{supp}X = \operatorname{supp}^\circ(\widetilde{X}) = \operatorname{Omb}(X) = \{Z\in\mathcal{M} \mid |Z|\subset|X|\} \]

\[ \operatorname{supp}\{P_a\} = \{P_a\} \]

\[ \operatorname{supp}\widetilde{S}_{a,b} = \left\{Z\in\mathcal{M} \;\middle|\;
\begin{array}{l} Z=P_x,\ a\leq x\leq b \\ Z=S_{a',b'},\ a\leq a'<b'\leq b \end{array}\right\} \]

\[ \operatorname{supp}^\circ(\widetilde{X}\setminus\alpha) = \operatorname{supp}X\setminus\alpha \]

\[ = \left\{Z\in\mathcal{M} \;\middle|\;
\begin{array}{l} Z=P_x,\ a\leq x\leq b \text{ et } x\notin\alpha \\ Z=S_{a',b'},\ a\leq a'<b'\leq b \end{array}\right\} \]

\[ \operatorname{supp}^\circ X = \operatorname{Omb}^\circ X = \{Z\in\mathcal{M} \mid Z \overset{\circ}{<} X\} \]

\[ \operatorname{supp}^\circ S_{a,b} = \left\{Z\in\mathcal{M} \;\middle|\;
\begin{array}{l} Z=P_x,\ a<x<b \\ Z=S_{a',b'},\ a\leq a'<b'\leq b \end{array}\right\} \]

\[ \operatorname{supp}^\circ(S_{a,b}\cup\beta) = \operatorname{supp}^\circ(S_{a,b})\cup\beta = \operatorname{Omb}^\circ(S_{a,b})\cup\beta \]

\[ = \left\{Z\in\mathcal{M} \;\middle|\;
\begin{array}{l} Z=P_x,\ a<x<b \text{ ou } x\in\beta \\ Z=S_{a',b'},\ a\leq a'<b'\leq b \end{array}\right\} . \]

\[ \operatorname{supp}\Phi = \operatorname{Omb}(\widetilde{S}_{a,b})\setminus\alpha = \operatorname{Omb}^\circ(S_{a,b})\cup\beta \]

\[ (\Phi = \Phi_{T,\alpha}) \qquad \text{(réunion disjointe)} \]

\[ \Phi \underset{\mathrm{pos}}{<} X < \Psi \]

\[ \operatorname{supp}^\circ(\Phi\cup\Psi) = \operatorname{supp}^\circ(\Phi)\cup\operatorname{supp}^\circ(\Psi) \]

\[ Z\in\operatorname{supp}^\circ(\Phi\cup\Psi) \overset{?}{\Longrightarrow} Z\in\operatorname{supp}^\circ\Phi \ \text{ou}\ Z\in\operatorname{supp}^\circ\Psi . \]

\[ \Phi \underset{\mathrm{pos}}{<} X \underset{\mathrm{pos}}{<} \Psi \]

\[ X\in\operatorname{cosupp}^\circ(\Phi\cup\Psi), \]

\[ Z \underset{\mathrm{pos}}{\leq} X \ \text{ou}\ Z \underset{\mathrm{pos}}{\geq} X . \]

\[ Z<X \Longrightarrow Z\in\operatorname{supp}^\circ(\Phi) \]

\[ Z>X \Longrightarrow Z\in\operatorname{supp}^\circ(\Psi) . \]

\[ Z<X \quad (\text{et } Z\in\operatorname{supp}^\circ(\Phi\cup\Psi)) \]

\[ Z'\in\operatorname{cosupp}^\circ(\Phi), \quad\text{i.e.}\quad Z'\,|{\circ}|\,\Phi \]

\[ \delta X = \{\operatorname{ex}\delta\Phi,\ \operatorname{or}\delta\Psi\} \quad (\operatorname{ex}\delta\Phi=a,\ \operatorname{or}\delta\Psi=b) \]

\[ \begin{cases} X=P_a & \text{si } a=b \\ X=S_{a,b} & \text{si } a<b \end{cases} \]

\[ Z' \underset{\mathrm{pos}}{<} P_a, \ \text{on aura}\ Z' \underset{\mathrm{pos}}{<} \Psi, \]

\[ \Phi = \{S_{a,b},\ P_b,\ S_{b,c}\} . \]

\[ \operatorname{supp}^\circ(\Phi) = \operatorname{supp}^\circ(S_{a,c}) \]

\[ X \underset{\mathrm{pos}}{<} \Phi \ (\text{i.e. } X<S_{a,b}) \iff X<S_{a,c} \]

\[ X \underset{\mathrm{pos}}{>} \Phi \ (\text{i.e. } X>S_{b,c}) \iff X>S_{a,c} \]

\[ F = F_D, \quad D = (T_1 \underset{\mathrm{pos}}{<} T_2 < \cdots \underset{\mathrm{pos}}{<} T_n) \in \operatorname{Drap}^*(\operatorname{Drap}^*(\mathcal{L})), \]

\[ \beta_i \subset \partial F_{T_i} = \partial T_i . \]

\[ \Phi = (F_D\setminus F_D\cap\mathcal{M}_0)\cup\bigcup\beta_i
= \bigcup_i\big((F_{T_i}\setminus F_{T_i}\cap\mathcal{M}_0)\cup\beta_i\big) \]

\[ \operatorname{supp}^\circ\Phi \overset{\text{th}}{=} \operatorname{Omb}^\circ(\Phi) \]

\[ \operatorname{supp}^\circ\Phi = \operatorname{supp}^\circ\Phi' \]

\[ \forall x\in\Phi'\cap\mathcal{M}_0, \quad \operatorname{card}\{X\in\Phi' \mid x\triangleleft X\}\ \text{\struck{$\neq 2$}} \]

\[ \operatorname{supp}^\circ(\Phi) = \operatorname{Omb}^\circ(\Phi) = \bigcup_{X\in\Phi}\operatorname{Omb}^\circ(X) \]

\[ \text{(réunion disjointe).} \]

\[ \Phi_0 = \Big(\bigcup_{X\in\Phi_1}\partial X\Big)\cap S \cup \Phi_{0,\mathrm{is}} \]

\[ \Phi_{0,\mathrm{is}} = \text{ens. des sommets isolés de } \Phi \]

\[ = \text{ens. des éléts de } S\cap\mathcal{M}_0 \text{ qui sont maximaux dans } S \text{ pour } \ll \]

\[ \overline{S} = \operatorname{supp}^\circ(\overline{\Phi}) = \operatorname{supp}(\overline{\Phi}) \]

\[ \text{\struck{$\overline{\Phi}$}}\qquad \Phi_1 = \overline{\Phi}\setminus\{\text{ens. des points isolés de } \overline{\Phi}\} \]

\[ \overline{\Phi}\setminus\Phi \subset \partial\Phi \quad \text{\struck{$\overline{\Phi}\setminus\Phi$}} = \{s\in\delta\Phi \mid \text{ordre de } s \text{ dans } \overline{\Phi} \text{ est } 1\} \]

\[ \Phi = \{P_a,\ S_{a,b},\ S_{b,c},\ P_c,\ S_{c,d},\ e\} \]

\[ \overline{\Phi} = \{P_a,\ S_{a,b},\ P_b,\ S_{b,c},\ P_c,\ S_{c,d},\ d,\ e\} \]

\[ \partial\Phi = \{a,\ d\} \]

\[ \delta\Phi \ (\text{ensemble des sommets de } \Phi) = \overline{\Phi}\cap\mathcal{L} = (\overline{\Phi})_0 \]

\[ \operatorname{ord}(a,\Phi) = \operatorname{card}(\{X\in\Phi_1 \mid a\triangleleft X\}) \in \{0,1,2\} \]

\[ \Phi_{\mathrm{is}} = \{s\in\delta\Phi \mid \operatorname{ord}(s,\Phi)=0\} \subset \Phi_0 \]

\[ \partial\Phi = \{s\in\delta\Phi \ (\text{ou } s\in\mathcal{L}) \mid \operatorname{ord}(s,\Phi)=1\} \]

\[ \Phi_0\cap\partial\Phi = \partial_0\Phi \subset \Phi \]

\[ \text{(sommets-bord propres)} \]

\[ \delta_{\mathrm{int}}(\Phi) = \{s\in\delta\Phi \ (\text{ou } s\in\mathcal{L}) \mid \operatorname{ord}(s,\Phi)=2\} \]

\[ \widetilde{S}_{a,b} = \{S_{a,b},\ a,\ b\} \]

\[ \Phi_0 = \Phi\cap\mathcal{M}_0 = \Phi\cap\mathcal{L}, \qquad \Phi_1 = \Phi\cap\mathcal{M}_1 \]

\[ \Phi \longmapsto \overline{\Phi} \longmapsto \overline{\Phi}\setminus\overline{\Phi}_{\mathrm{is}} \Longrightarrow \Phi_{\mathrm{comp}} = \Psi \]

\[ D \in \operatorname{Drap}^*(\mathcal{M}_1,\overset{1}{<}) \]

\[ D = \{X_1,\ X_2,\ \ldots,\ X_n\} \]

\[ X_i = S_{a_i,b_i}, \]

\[ \begin{cases}
\Psi = \Phi_{\mathrm{comp}} = \Phi_D = \Phi_{X_1,X_2,\ldots,X_n} \\
\delta\Psi = \partial\Psi = \partial\Phi = \{a_1,b_1,a_2,b_2,\ldots,a_n,b_n\} .
\end{cases} \]

\[ \delta\Phi \in \operatorname{Drap}^*(\mathcal{L}) \supset \delta\Psi \]

\[
  \Delta_{\mathrm{int}} = (\Delta \setminus \partial\Psi) \cap \mathrm{Omb}(\Psi) = \Delta \cap \mathrm{Omb}^\circ(\Psi)
\]

\[
  \Delta_{\mathrm{is}} = \text{\struck{$\Delta \setminus (\partial\Psi \cup \Delta$}}\; (\Delta \setminus \partial\Psi) \setminus \Delta_{\mathrm{int}}
\]

\[
  \Delta = \partial\Psi \amalg \Delta_{\mathrm{int}} \amalg \Delta_{\mathrm{is}}
\]

\[
  \boxed{\Delta_{\mathrm{lac}}} \subset \Delta_{\mathrm{int}}
\]

\[
  \Delta_{\mathrm{int}} = \Delta_{\mathrm{lac}} \amalg \Delta_{\mathrm{red}} .
\]

\[
  \partial_0 \subset \partial\Psi .
\]

\[
  D = (X_1 \mathrel{<\!|} X_2 \mathrel{<\!|} \text{\struck{$X_3$}} \cdots X_n) \in \mathrm{Drap}^*(\mathcal{M}_1) .
\]

\[
  \Psi = \Phi_D = \bigcup_{X \in D} \widetilde{X}_{\xi} = \bigcup_{1 \leq i \leq n} \widetilde{X}_i
\]

\[
  \partial\Psi = \delta\Psi = \bigcup_{X \in D} \partial X = \bigcup_{1 \leq i \leq n} \partial X_i
\]

\[
  \delta\Psi = \{a_1, b_1, a_2, b_2, \ldots, a_n, b_n\}
\]

\[
  \partial X_i = \{a_i, b_i\} .
\]

\[
  a_1 < b_1 < a_2 < b_2 \cdots < a_n < b_n ,
\]

\[
  X_i = S_{a_i, b_i} \qquad 1 \leq i \leq n .
\]

\[
  \Delta \supset \partial\Psi
\]

\[
  \Delta_{\mathrm{int}} = \Delta \cap \mathrm{Omb}^\circ(\Psi_1) \subset \Delta \setminus \partial\Psi
\]

\[
  \Delta = \Delta_{\mathrm{is}} \amalg \Delta_{\mathrm{int}} \amalg \partial\Psi
\]

\[
  \Delta_{\mathrm{is}} \overset{\mathrm{def}}{=} (\Delta \setminus \partial\Psi) \setminus \Delta_{\mathrm{int}}
\]

\[
  \Delta_{\mathrm{lac}} \subset \Delta_{\mathrm{int}}
\]

\[
  \Delta_{\mathrm{int}} = \Delta_{\mathrm{lac}} \amalg \Delta_{\mathrm{red}}
\]

\[
  \Delta_{\mathrm{red}} \overset{\mathrm{def}}{=} \Delta_{\mathrm{int}} \setminus \Delta_{\mathrm{lac}} .
\]

\[
  \partial_0 \subset \partial\Psi
\]

\[
  \begin{cases}
    \Phi_0 = \underbrace{\partial_0}_{\text{sommets bord propres}} \cup \underbrace{\Delta_{\mathrm{red}}}_{\text{sommets redondants}} \cup \Delta_{\mathrm{is}} \\[1ex]
    \Phi_1 = \{ X \in \underbrace{\Phi_\Delta \cap \mathcal{M}_1}_{\text{intervalles interstitiels de } \Delta} \mid X \in \mathrm{Omb}(\Psi) \}
  \end{cases}
\]

\[
  \begin{gathered}
    \delta\Phi = \Delta \\
    \partial\Phi = \partial\Psi , \quad \partial_0\Phi = \partial_0 , \quad \partial_{\mathrm{imp}}\Phi = \partial\Psi \setminus \partial_0 \\
    \Phi_{\mathrm{is}} = \Delta_{\mathrm{is}} \\
    \Phi_{\mathrm{red}} = \Delta_{\mathrm{red}} \\
    \Phi_{\mathrm{lac}} = \Delta_{\mathrm{lac}}
  \end{gathered}
\]

\[
  \Psi \quad \text{ou} \quad X_1 \mathrel{<\!|} X_2 \mathrel{<\!|} X_3 \cdots \mathrel{<\!|} X_n
\]

\[
  \text{ou} \quad a_1 < b_1 < a_2 < b_2 \cdots < a_n < b_n
\]

\[
  \boxed{\partial_0} \subset \partial\Psi \subset \boxed{\Delta} \supset \Delta_{\mathrm{lac}}
\]

\[
  \Delta \in \mathrm{Drap}^*(\mathcal{L}) ,
\]

\[
  \Delta_{\mathrm{lac}} \subset \mathrm{Omb}^\circ(\Psi_1)\, \text{\struck{\ill{}}} \cap \Delta
  \quad \bigl(= \mathrm{Omb}(\Psi_1) \cap (\Delta \setminus \partial\Psi)\bigr) .
\]

\[
  \Delta_{\mathrm{red}} = \emptyset
\]

\[
  \Delta_{\mathrm{lac}} = \Delta \cap \mathrm{Omb}^\circ(\Psi_1) ,
\]

\[
  \partial_0 \subset \partial\Psi \subset \Delta
\]

\[
  \mathrm{Supp}^\circ(\Phi) = \mathrm{Omb}^\circ(\Phi_1) \cup \text{\struck{\ill{}\,$\Delta_{\mathrm{is}}$}}\; \Phi_0
\]

\[
  \Phi'^*_0 = \underbrace{\delta\Phi}_{\Delta} \setminus \Phi_0 = \Delta \setminus \Phi_0 = \Delta_{\mathrm{lac}} \amalg \partial'_0
\]

\[
  \partial'_0 \overset{\mathrm{def}}{=} \partial\Phi \setminus \partial_0\Phi = \partial\Psi \setminus \partial_0
\]

\[
  \Phi'^*_1 = \text{\struck{$\{ X \in \Phi_\Delta \cap \mathcal{M}_1 \mid X \notin \Phi_1$}\,\struck{\ill{}}}\; \}
\]

\[
  \text{\uncertain{Pur}\ill{}}_\Delta : \quad \forall s \in \Delta \setminus \Phi_0 ,\ \exists X \in \Phi_1 \text{ t.q. } s \lhd X ,
\]

\[
  \Delta = \Phi_0 \cup \bigcup_{X \in \Phi_1} \partial X .
\]

\[
  \Phi'^*_0 = \text{\struck{$\Delta \setminus \Phi$}}\; (\Phi_\Delta)_0 \setminus \Phi_0 = \Delta \setminus \Phi_0
\]

\[
  \Phi'^*_1 = (\Phi_\Delta)_1 \setminus \Phi_1
\]

\[
  a = \mathrm{or}(\Delta) , \quad b = \mathrm{ex}(\Delta)
\]

\[
  \begin{gathered}
    \mathcal{M}_{<a} = \{ X \in \mathcal{M} \mid X < a \} \\
    \mathcal{M}_{>b} = \{ X \in \mathcal{M} \mid X > b \}
  \end{gathered}
\]

\[
  \mathrm{Cosupp}^\circ(\Phi) = \mathrm{Omb}^\circ(\Phi'^*_1) \cup \mathcal{M}_{<a} \cup \mathcal{M}_{>b}
\]

\[
  \mathrm{Omb}^\circ(\Phi'^*) \overset{\mathrm{def}}{=} \bigcup_{X \in \Phi'^*} \mathrm{Omb}^\circ(X) \qquad \text{réunion disjointe}
\]