Cote n° 156-6 · pages 1–92
· 181 displayed formulas · [Chapitre] VI. Analysis situs (deuxième mouture) : notes manuscrites (18-20/06/1986).
Inventory dating : 1986
Édition de démonstration
\[(1.1.) \qquad \mathcal{L},\ \mathcal{M},\ \mathfrak{F} ,\]
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\[
(1.1.) \qquad \mathcal{L},\ \mathcal{M},\ \mathfrak{F} ,
\]\[(1.2) \qquad \mathcal{L} \overset{b)}{\hookrightarrow} \mathcal{M} \overset{a)}{\hookrightarrow} \mathfrak{F} ,\]
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\[
(1.2) \qquad \mathcal{L} \overset{b)}{\hookrightarrow} \mathcal{M} \overset{a)}{\hookrightarrow} \mathfrak{F} ,
\]\[(1.3) \qquad
\begin{cases}
a)\ \mathfrak{F} \hookrightarrow \mathfrak{P}(\mathcal{M}) & F \mapsto \widetilde{F} = \mathrm{Dépl}(F) \subset \mathcal{M} \\
b)\ \mathfrak{F} \hookrightarrow \mathfrak{P}(\mathfrak{P}(\mathcal{M})) & \text{plus précisément } \mathfrak{F} \to \mathrm{Fig}(\mathcal{M}), \\
& F \mapsto \mathrm{Fig}_{\mathcal{M}}(F) \subset \mathfrak{P}(\mathcal{M})
\end{cases}\]
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\[
(1.3) \qquad
\begin{cases}
a)\ \mathfrak{F} \hookrightarrow \mathfrak{P}(\mathcal{M}) & F \mapsto \widetilde{F} = \mathrm{Dépl}(F) \subset \mathcal{M} \\
b)\ \mathfrak{F} \hookrightarrow \mathfrak{P}(\mathfrak{P}(\mathcal{M})) & \text{plus précisément } \mathfrak{F} \to \mathrm{Fig}(\mathcal{M}), \\
& F \mapsto \mathrm{Fig}_{\mathcal{M}}(F) \subset \mathfrak{P}(\mathcal{M})
\end{cases}
\]\[(1.4) \qquad \mathrm{Fig}_{\mathcal{M}}(F) = \lbrace \widetilde{X} \mid X \in \widetilde{F} = \mathrm{Dépl}(F) \rbrace \subset \mathfrak{P}(\mathcal{M})
\quad \text{i.e.} \quad \mathrm{Fig}_{\mathcal{M}}(F) \in \mathfrak{P}(\mathfrak{P}(\mathcal{M}))\]
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\[
(1.4) \qquad \mathrm{Fig}_{\mathcal{M}}(F) = \lbrace \widetilde{X} \mid X \in \widetilde{F} = \mathrm{Dépl}(F) \rbrace \subset \mathfrak{P}(\mathcal{M})
\quad \text{i.e.} \quad \mathrm{Fig}_{\mathcal{M}}(F) \in \mathfrak{P}(\mathfrak{P}(\mathcal{M}))
\]\[(1.5) \qquad \widetilde{X} = \lbrace Y \in \mathcal{M} \mid Y \leq X \rbrace .\]
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\[
(1.5) \qquad \widetilde{X} = \lbrace Y \in \mathcal{M} \mid Y \leq X \rbrace .
\]\[X \longmapsto \widetilde{X} = \lbrace Y \in \mathcal{M} \mid Y \leq X \rbrace \qquad \mathcal{M} \to \mathfrak{P}_{\mathrm{ferm}}(\mathcal{M}) .\]
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\[
X \longmapsto \widetilde{X} = \lbrace Y \in \mathcal{M} \mid Y \leq X \rbrace \qquad \mathcal{M} \to \mathfrak{P}_{\mathrm{ferm}}(\mathcal{M}) .
\]\[(1.6) \qquad
\begin{cases}
a)\ \mathfrak{F} \to \mathfrak{P}(\mathcal{M}) & F \mapsto \mathrm{Omb}_{\mathcal{M}}(F) \\
b)\ \mathfrak{F} \hookrightarrow \mathfrak{P}\mathfrak{P}(\mathcal{M}) \ \text{et} = \mathfrak{F} \to \mathrm{Fig}(\mathcal{M}) \subset \mathfrak{P}\mathfrak{P}(\mathcal{M}) & F \mapsto \mathrm{Multomb}_{\mathcal{M}}(F) ,
\end{cases}\]
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\[
(1.6) \qquad
\begin{cases}
a)\ \mathfrak{F} \to \mathfrak{P}(\mathcal{M}) & F \mapsto \mathrm{Omb}_{\mathcal{M}}(F) \\
b)\ \mathfrak{F} \hookrightarrow \mathfrak{P}\mathfrak{P}(\mathcal{M}) \ \text{et} = \mathfrak{F} \to \mathrm{Fig}(\mathcal{M}) \subset \mathfrak{P}\mathfrak{P}(\mathcal{M}) & F \mapsto \mathrm{Multomb}_{\mathcal{M}}(F) ,
\end{cases}
\]\[(1.7) \qquad \mathrm{Multomb}(F) = \lbrace X \in \widetilde{F} \mid \mathrm{Omb}\, X \rbrace ,\]
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\[
(1.7) \qquad \mathrm{Multomb}(F) = \lbrace X \in \widetilde{F} \mid \mathrm{Omb}\, X \rbrace ,
\]\[(1.8) \qquad \mathrm{Multomb}(X) = \lbrace Y \in \mathcal{M} \mid Y \ll X \rbrace ,\]
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\[
(1.8) \qquad \mathrm{Multomb}(X) = \lbrace Y \in \mathcal{M} \mid Y \ll X \rbrace ,
\]\[\mathcal{L} \hookrightarrow \mathcal{M} ,\]
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\[
\mathcal{L} \hookrightarrow \mathcal{M} ,
\]\[\mathfrak{P}(\mathcal{M}) \to \mathfrak{P}(\mathcal{L}) , \qquad \mathrm{Fig}(\mathcal{M}) \to \mathrm{Fig}(\mathcal{L}) ,\]
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\[
\mathfrak{P}(\mathcal{M}) \to \mathfrak{P}(\mathcal{L}) , \qquad \mathrm{Fig}(\mathcal{M}) \to \mathrm{Fig}(\mathcal{L}) ,
\]\[(1.9) \qquad
\begin{cases}
a)\ \mathfrak{F} \to \mathfrak{P}(\mathcal{L}) & F \mapsto \mathrm{omb}(F) \\
b)\ \mathfrak{F} \to \mathrm{Fig}(\mathcal{L}) & F \mapsto \mathrm{multomb}(F)
\end{cases} ,\]
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\[
(1.9) \qquad
\begin{cases}
a)\ \mathfrak{F} \to \mathfrak{P}(\mathcal{L}) & F \mapsto \mathrm{omb}(F) \\
b)\ \mathfrak{F} \to \mathrm{Fig}(\mathcal{L}) & F \mapsto \mathrm{multomb}(F)
\end{cases} ,
\]\[(1.10) \qquad \leq,\ \ll,\ \preccurlyeq \quad \text{relations d'ordre dans } \mathfrak{F}\]
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\[
(1.10) \qquad \leq,\ \ll,\ \preccurlyeq \quad \text{relations d'ordre dans } \mathfrak{F}
\]\[(1.11) \qquad \leq,\ \ll \quad \text{dans } \mathcal{M}\]
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\[
(1.11) \qquad \leq,\ \ll \quad \text{dans } \mathcal{M}
\]\[(1.12) \qquad
\begin{cases}
F \text{ et } G \ \underline{\text{compatibles}} \\
F \text{ et } G \ \underline{\text{disjoints}}
\end{cases} ,\]
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\[
(1.12) \qquad
\begin{cases}
F \text{ et } G \ \underline{\text{compatibles}} \\
F \text{ et } G \ \underline{\text{disjoints}}
\end{cases} ,
\]\[F \text{ et } G \text{ disjoints} \Longrightarrow F \text{ et } G \text{ compatibles.}\]
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\[
F \text{ et } G \text{ disjoints} \Longrightarrow F \text{ et } G \text{ compatibles.}
\]\[(1.13) \qquad F \text{ et } G \text{ disjoints} \iff\]
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\[
(1.13) \qquad F \text{ et } G \text{ disjoints} \iff
\]\[(1.14) \qquad F \parallel G\]
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\[ (1.14) \qquad F \parallel G \]
\[(1.15) \qquad F \mathbin{\Diamond} G \quad \text{ou} \quad F \mathbin{\between} G\]
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\[
(1.15) \qquad F \mathbin{\Diamond} G \quad \text{ou} \quad F \mathbin{\between} G
\]\[(1.16) \qquad
\begin{cases}
a)\ \leq,\ \ll \\
b)\ \leq,\ \preccurlyeq \\
c)\ \ll,\ \text{et}\ \mathfrak{F} \to \mathrm{P}(\mathfrak{F})
\end{cases}\]
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\[
(1.16) \qquad
\begin{cases}
a)\ \leq,\ \ll \\
b)\ \leq,\ \preccurlyeq \\
c)\ \ll,\ \text{et}\ \mathfrak{F} \to \mathrm{P}(\mathfrak{F})
\end{cases}
\]\[(1.17) \qquad F \mathrel{\triangleleft} G \overset{\mathrm{dfn}}{\iff} \text{« $F$ incident à $G$ » i.e. } F \leq G \text{ et } F \in \mathcal{M}\]
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\[
(1.17) \qquad F \mathrel{\triangleleft} G \overset{\mathrm{dfn}}{\iff} \text{« $F$ incident à $G$ » i.e. } F \leq G \text{ et } F \in \mathcal{M}
\]\[(1.19) \qquad F \parallel F \iff F = \emptyset_{\mathfrak{F}} \quad \text{figure vide.}\]
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\[
(1.19) \qquad F \parallel F \iff F = \emptyset_{\mathfrak{F}} \quad \text{figure vide.}
\]\[(1.20) \qquad \mathcal{M} = \lbrace X \in \mathfrak{F} \mid X \text{ strict.\ irréductible pour } \leq ,\ \text{i.e. } X \leq \textstyle\sup_i F_i \Rightarrow \exists\, i \text{ avec } X \leq F_i \rbrace\]
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\[
(1.20) \qquad \mathcal{M} = \lbrace X \in \mathfrak{F} \mid X \text{ strict.\ irréductible pour } \leq ,\ \text{i.e. } X \leq \textstyle\sup_i F_i \Rightarrow \exists\, i \text{ avec } X \leq F_i \rbrace
\]\[(1.21) \qquad \mathcal{M} = \lbrace X \in \mathfrak{F} \mid X \mathrel{\triangleleft} X \rbrace\]
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\[
(1.21) \qquad \mathcal{M} = \lbrace X \in \mathfrak{F} \mid X \mathrel{\triangleleft} X \rbrace
\]\[(1.22) \qquad \widetilde{F} = \lbrace X \in \mathcal{M} \mid X \mathrel{\triangleleft} F \rbrace\]
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\[
(1.22) \qquad \widetilde{F} = \lbrace X \in \mathcal{M} \mid X \mathrel{\triangleleft} F \rbrace
\]\[(1.23) \qquad F \leq G \iff \widetilde{F} \subset \widetilde{G}\]
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\[
(1.23) \qquad F \leq G \iff \widetilde{F} \subset \widetilde{G}
\]\[(1.24) \qquad
\begin{cases}
a)\ F \parallel G \iff \lbrace F, G \rbrace \text{ majoré pour } \leq, \text{ et } \mathrm{Inf}_{\leq}(F,G) = \emptyset_{\mathfrak{F}} \\
b)\ F \parallel G \iff \mathrm{Sup}_{\ll}(F,G) \text{ existe, et } \mathrm{Inf}_{\ll}\, F,G = \emptyset_{\mathfrak{F}}
\end{cases}\]
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\[
(1.24) \qquad
\begin{cases}
a)\ F \parallel G \iff \lbrace F, G \rbrace \text{ majoré pour } \leq, \text{ et } \mathrm{Inf}_{\leq}(F,G) = \emptyset_{\mathfrak{F}} \\
b)\ F \parallel G \iff \mathrm{Sup}_{\ll}(F,G) \text{ existe, et } \mathrm{Inf}_{\ll}\, F,G = \emptyset_{\mathfrak{F}}
\end{cases}
\]\[(1.25) \qquad F \preccurlyeq G \iff F \ll G, \text{ et } \forall H \in \mathfrak{F} \text{ on a } (H \parallel F \iff H \parallel G)\]
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\[
(1.25) \qquad F \preccurlyeq G \iff F \ll G, \text{ et } \forall H \in \mathfrak{F} \text{ on a } (H \parallel F \iff H \parallel G)
\]\[(1.26) \qquad \mathcal{L} = \lbrace X \in \mathfrak{F},\ X \neq \emptyset_{\mathfrak{F}} \mid Y \in \mathfrak{F},\ Y \ll X,\ Y \neq \emptyset_{\mathfrak{F}} \Rightarrow Y = X \rbrace\]
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\[
(1.26) \qquad \mathcal{L} = \lbrace X \in \mathfrak{F},\ X \neq \emptyset_{\mathfrak{F}} \mid Y \in \mathfrak{F},\ Y \ll X,\ Y \neq \emptyset_{\mathfrak{F}} \Rightarrow Y = X \rbrace
\]\[(1.27) \qquad G \leq F \iff (G \ll F \text{ et } G \between F)\]
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\[
(1.27) \qquad G \leq F \iff (G \ll F \text{ et } G \between F)
\]\[(1.28) \qquad
\begin{cases}
\text{soit par } \boxed{\leq,\ \ll} \\
\text{soit par } \leq,\ \preccurlyeq \\
\text{soit par } \ll,\ \between .
\end{cases}\]
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\[
(1.28) \qquad
\begin{cases}
\text{soit par } \boxed{\leq,\ \ll} \\
\text{soit par } \leq,\ \preccurlyeq \\
\text{soit par } \ll,\ \between .
\end{cases}
\]\[(1.29) \qquad G \ll F \iff \exists\, F' \text{ avec } G \leq F' \preccurlyeq F\]
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\[
(1.29) \qquad G \ll F \iff \exists\, F' \text{ avec } G \leq F' \preccurlyeq F
\]\[(1.30) \qquad \boxed{\leq,\ \ll \ \text{(rel.\ d'ordre dans } \mathfrak{F}) \ \text{telles que} \ F \leq G \Rightarrow F \ll G}\]
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\[
(1.30) \qquad \boxed{\leq,\ \ll \ \text{(rel.\ d'ordre dans } \mathfrak{F}) \ \text{telles que} \ F \leq G \Rightarrow F \ll G}
\]\[\mathcal{M} \subset \mathfrak{F}\]
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\[
\mathcal{M} \subset \mathfrak{F}
\]\[\mathcal{L} \subset \mathcal{M}\]
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\[
\mathcal{L} \subset \mathcal{M}
\]\[(*) \qquad \mathfrak{F} \hookrightarrow \mathfrak{P}(\mathcal{M}) \qquad F \mapsto \widetilde{F} = \mathrm{Dépl}(F)\]
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\[
(*) \qquad \mathfrak{F} \hookrightarrow \mathfrak{P}(\mathcal{M}) \qquad F \mapsto \widetilde{F} = \mathrm{Dépl}(F)
\]\[(1.31) \qquad \widetilde{F} = \lbrace X \in \mathcal{M} \mid X \leq F \rbrace\]
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\[
(1.31) \qquad \widetilde{F} = \lbrace X \in \mathcal{M} \mid X \leq F \rbrace
\]\[(**) \qquad \mathfrak{F} \hookrightarrow \mathrm{Fig}(\mathcal{M}) \qquad F \mapsto \mathrm{Fig}_{\mathcal{M}}(F)\]
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\[
(**) \qquad \mathfrak{F} \hookrightarrow \mathrm{Fig}(\mathcal{M}) \qquad F \mapsto \mathrm{Fig}_{\mathcal{M}}(F)
\]\[\mathrm{Fig}_{\mathcal{M}}(F) = \lbrace \widetilde{X} \mid X \in \widetilde{F} \rbrace \subset \mathfrak{P}(\mathcal{M})
\quad \text{avec} \quad \widetilde{X} = \lbrace Y \in \mathcal{M} \mid Y \leq X \rbrace .\]
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\[
\mathrm{Fig}_{\mathcal{M}}(F) = \lbrace \widetilde{X} \mid X \in \widetilde{F} \rbrace \subset \mathfrak{P}(\mathcal{M})
\quad \text{avec} \quad \widetilde{X} = \lbrace Y \in \mathcal{M} \mid Y \leq X \rbrace .
\]\[(1.32) \qquad F \leq G \iff \widetilde{F} \subset \widetilde{G}\]
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\[
(1.32) \qquad F \leq G \iff \widetilde{F} \subset \widetilde{G}
\]\[(1.33) \qquad
\boxed{
\begin{array}{l}
\mathfrak{F} \subset \mathfrak{P}(\mathcal{M}) \ \text{ens.\ de parties de } \mathcal{M} \\
\ll \ \text{relation d'ordre dans } \mathfrak{F}, \text{ impliquée par l'inclusion (notée } F \leq G)
\end{array}}\]
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\[
(1.33) \qquad
\boxed{
\begin{array}{l}
\mathfrak{F} \subset \mathfrak{P}(\mathcal{M}) \ \text{ens.\ de parties de } \mathcal{M} \\
\ll \ \text{relation d'ordre dans } \mathfrak{F}, \text{ impliquée par l'inclusion (notée } F \leq G)
\end{array}}
\]\[(1.34) \qquad F \in \mathfrak{F} \text{ de t.f.} \iff \widetilde{F} = \text{réunion finie d'ens.\ } \mathcal{M}_{\leq X_i}\]
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\[
(1.34) \qquad F \in \mathfrak{F} \text{ de t.f.} \iff \widetilde{F} = \text{réunion finie d'ens.\ } \mathcal{M}_{\leq X_i}
\]\[(1.35) \qquad \mathbin{\between_{\mathcal{M}}} \ \text{relation binaire sym.\ réfl.\ (sur } \mathcal{M})\]
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\[
(1.35) \qquad \mathbin{\between_{\mathcal{M}}} \ \text{relation binaire sym.\ réfl.\ (sur } \mathcal{M})
\]\[(1.36) \qquad X \between Y,\quad X' \leq X,\ Y' \leq Y \Longrightarrow X' \between Y'\]
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\[ (1.36) \qquad X \between Y,\quad X' \leq X,\ Y' \leq Y \Longrightarrow X' \between Y' \]
\[(*) \qquad \mathrm{Omb}(F) = \lbrace X \in \mathcal{M} \mid X \ll F \rbrace\]
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\[
(*) \qquad \mathrm{Omb}(F) = \lbrace X \in \mathcal{M} \mid X \ll F \rbrace
\]\[(**) \qquad \mathrm{Multomb}(F) = \lbrace \mathrm{Omb}(X) \mid X \in \widetilde{F} \rbrace .\]
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\[
(**) \qquad \mathrm{Multomb}(F) = \lbrace \mathrm{Omb}(X) \mid X \in \widetilde{F} \rbrace .
\]\[(*) \qquad \mathfrak{F} \longrightarrow \mathrm{Fig}(\mathcal{M}) , \qquad F \longmapsto \mathrm{Multomb}(F)\]
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\[
(*) \qquad \mathfrak{F} \longrightarrow \mathrm{Fig}(\mathcal{M}) , \qquad F \longmapsto \mathrm{Multomb}(F)
\]\[(1.38) \qquad G \leq F \iff \mathrm{Multomb}(G) \leq \mathrm{Multomb}(F)\]
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\[
(1.38) \qquad G \leq F \iff \mathrm{Multomb}(G) \leq \mathrm{Multomb}(F)
\]\[F^{X} = \lbrace Y \in \widetilde{F} \mid \underbrace{X \ll Y}_{X \in \mathrm{Omb}(Y)} \rbrace\]
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\[
F^{X} = \lbrace Y \in \widetilde{F} \mid \underbrace{X \ll Y}_{X \in \mathrm{Omb}(Y)} \rbrace
\]\[\mathrm{Multomb}(F) = \lbrace \widetilde{Y} \mid Y \in \widetilde{F}, \ Y \trianglelefteq F \rbrace\]
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\[
\mathrm{Multomb}(F) = \lbrace \widetilde{Y} \mid Y \in \widetilde{F}, \ Y \trianglelefteq F \rbrace
\]\[\widetilde{Y}^{\circ} \overset{\mathrm{def}}{=} \widetilde{Y} \smallsetminus \bigcup_{\substack{Y' \in \widetilde{F} \\ Y' \leq Y}} \widetilde{Y}'\]
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\[
\widetilde{Y}^{\circ} \overset{\mathrm{def}}{=} \widetilde{Y} \smallsetminus \bigcup_{\substack{Y' \in \widetilde{F} \\ Y' \leq Y}} \widetilde{Y}'
\]\[\mathcal{M} \hookrightarrow \mathfrak{P}(\mathcal{M}), \qquad X \longmapsto \mathrm{Omb}(X) = \mathcal{M}_{\ll X} = \lbrace Y \in \mathcal{M} \mid Y \ll X \rbrace \tag{1.39}\]
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\[
\mathcal{M} \hookrightarrow \mathfrak{P}(\mathcal{M}), \qquad X \longmapsto \mathrm{Omb}(X) = \mathcal{M}_{\ll X} = \lbrace Y \in \mathcal{M} \mid Y \ll X \rbrace \tag{1.39}
\]\[(*) \qquad \mathrm{Multomb}(F) \between \mathrm{Multomb}(G),\]
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\[
(*) \qquad \mathrm{Multomb}(F) \between \mathrm{Multomb}(G),
\]\[(**) \qquad \begin{cases} \mathrm{Multomb}(F \vee G) = \mathrm{Multomb}(F) \cup \mathrm{Multomb}(G) \\ \mathrm{Multomb}(F \wedge G) = \mathrm{Multomb}(F) \cap \mathrm{Multomb}(G). \end{cases}\]
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\[
(**) \qquad \begin{cases} \mathrm{Multomb}(F \vee G) = \mathrm{Multomb}(F) \cup \mathrm{Multomb}(G) \\ \mathrm{Multomb}(F \wedge G) = \mathrm{Multomb}(F) \cap \mathrm{Multomb}(G). \end{cases}
\]\[\mathrm{Multomb}(F \vee G) = \mathrm{Multomb}(F) \cup \mathrm{Multomb}(G),\]
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\[
\mathrm{Multomb}(F \vee G) = \mathrm{Multomb}(F) \cup \mathrm{Multomb}(G),
\]\[\mathrm{Multomb}(\emptyset_{\mathfrak{F}}) = \emptyset \quad (\text{figure vide dans } \mathcal{M}) \tag{1.40}\]
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\[
\mathrm{Multomb}(\emptyset_{\mathfrak{F}}) = \emptyset \quad (\text{figure vide dans } \mathcal{M}) \tag{1.40}
\]\[\mathrm{Omb}(F) = |\mathrm{Multomb}\, F| \overset{\mathrm{def}}{=} \bigcup_{X \in \widetilde{F}} \mathrm{Omb}(X)\]
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\[
\mathrm{Omb}(F) = |\mathrm{Multomb}\, F| \overset{\mathrm{def}}{=} \bigcup_{X \in \widetilde{F}} \mathrm{Omb}(X)
\]\[\mathrm{Omb}(F) \xrightarrow{\ \varphi_F\ } \widetilde{F}\]
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\[
\mathrm{Omb}(F) \xrightarrow{\ \varphi_F\ } \widetilde{F}
\]\[|\Psi| \subset |\Phi| \qquad \text{i.e.} \qquad \mathrm{Omb}\, G \subset \mathrm{Omb}\, F,\]
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\[
|\Psi| \subset |\Phi| \qquad \text{i.e.} \qquad \mathrm{Omb}\, G \subset \mathrm{Omb}\, F,
\]\[f : \widetilde{G} \to \widetilde{F}\]
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\[
f : \widetilde{G} \to \widetilde{F}
\]\[X \ll f(X),\]
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\[ X \ll f(X), \]
\[f : \widetilde{G} \to \widetilde{F}\]
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\[
f : \widetilde{G} \to \widetilde{F}
\]\[X' \leq X \ \text{ i.e. } \ X' \ll X \ (\ll f(X)) \ \text{ implique que } \ X' \ll f(X), \ \text{ donc } \ f(X') \ll f(X),\]
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\[
X' \leq X \ \text{ i.e. } \ X' \ll X \ (\ll f(X)) \ \text{ implique que } \ X' \ll f(X), \ \text{ donc } \ f(X') \ll f(X),
\]\[X \mathrel{\overset{\circ}{\ll}} Y \tag{1.43}\]
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\[
X \mathrel{\overset{\circ}{\ll}} Y \tag{1.43}
\]\[X \in \mathrm{Omb}(Y)^{\circ} = \mathrm{Omb}(Y) \smallsetminus \bigcup_{Y' < Y} \mathrm{Omb}(Y')\]
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\[
X \in \mathrm{Omb}(Y)^{\circ} = \mathrm{Omb}(Y) \smallsetminus \bigcup_{Y' < Y} \mathrm{Omb}(Y')
\]\[X \mathrel{\overset{\circ}{\ll}} Y \mathrel{\overset{\circ}{\ll}} Z \quad \text{alors} \quad X \mathrel{\overset{\circ}{\ll}} Z\]
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\[
X \mathrel{\overset{\circ}{\ll}} Y \mathrel{\overset{\circ}{\ll}} Z \quad \text{alors} \quad X \mathrel{\overset{\circ}{\ll}} Z
\]\[\mathfrak{F}_{\mathcal{M}} \subset \mathrm{Fig}(\mathcal{M}) \subset \mathfrak{P}(\mathfrak{P}(\mathcal{M})) \tag{1.44}\]
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\[
\mathfrak{F}_{\mathcal{M}} \subset \mathrm{Fig}(\mathcal{M}) \subset \mathfrak{P}(\mathfrak{P}(\mathcal{M})) \tag{1.44}
\]\[\mathcal{M}^{*} = \bigcup_{\Phi \in \mathfrak{F}_{\mathcal{M}}} \Phi \subset \mathfrak{P}(\mathcal{M}) \quad = \quad \lbrace A \in \mathfrak{P}(\mathcal{M}) \mid \exists\, \Phi \in \mathfrak{F}_{\mathcal{M}} \text{ avec } A \in \Phi \rbrace\]
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\[
\mathcal{M}^{*} = \bigcup_{\Phi \in \mathfrak{F}_{\mathcal{M}}} \Phi \subset \mathfrak{P}(\mathcal{M}) \quad = \quad \lbrace A \in \mathfrak{P}(\mathcal{M}) \mid \exists\, \Phi \in \mathfrak{F}_{\mathcal{M}} \text{ avec } A \in \Phi \rbrace
\]\[X, Y \in \mathcal{M},\ X \neq Y \Longrightarrow \exists\, A \in \mathcal{M}^{*} \text{ avec } X \in A,\ Y \notin A \ \text{ ou } \ Y \in A,\ X \notin A\]
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\[
X, Y \in \mathcal{M},\ X \neq Y \Longrightarrow \exists\, A \in \mathcal{M}^{*} \text{ avec } X \in A,\ Y \notin A \ \text{ ou } \ Y \in A,\ X \notin A
\]\[X \longmapsto \mathcal{M}_{\ll X} \overset{\mathrm{def}}{=} \mathrm{Omb}(X), \qquad (*) \quad \mathcal{M} \longrightarrow \mathfrak{P}(\mathcal{M})\]
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\[
X \longmapsto \mathcal{M}_{\ll X} \overset{\mathrm{def}}{=} \mathrm{Omb}(X), \qquad (*) \quad \mathcal{M} \longrightarrow \mathfrak{P}(\mathcal{M})
\]\[\mathcal{M} \xrightarrow{\ \sim\ } \mathcal{M}^{*} \subset \mathfrak{P}(\mathcal{M})\]
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\[
\mathcal{M} \xrightarrow{\ \sim\ } \mathcal{M}^{*} \subset \mathfrak{P}(\mathcal{M})
\]\[\Phi = \lbrace \mathrm{Omb}(X) \mid X \in F \rbrace, \qquad \mathrm{Omb}(X) = \mathcal{M}_{\ll X}\]
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\[
\Phi = \lbrace \mathrm{Omb}(X) \mid X \in F \rbrace, \qquad \mathrm{Omb}(X) = \mathcal{M}_{\ll X}
\]\[X \leq Y \Longleftrightarrow \Phi_{\mathrm{Omb}(X)} \subset \Phi_{\mathrm{Omb}(Y)}, \quad \text{i.e.} \quad \mathrm{Omb}(X) \in \Phi_{\mathrm{Omb}(Y)} \ \overset{\mathrm{def}}{=}\ \mathrm{Multomb}(Y)\]
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\[
X \leq Y \Longleftrightarrow \Phi_{\mathrm{Omb}(X)} \subset \Phi_{\mathrm{Omb}(Y)}, \quad \text{i.e.} \quad \mathrm{Omb}(X) \in \Phi_{\mathrm{Omb}(Y)} \ \overset{\mathrm{def}}{=}\ \mathrm{Multomb}(Y)
\]\[\widetilde{X} = \lbrace Y \in \mathcal{M} \mid Y \leq X \rbrace = \mathcal{M}_{\leq X}\]
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\[
\widetilde{X} = \lbrace Y \in \mathcal{M} \mid Y \leq X \rbrace = \mathcal{M}_{\leq X}
\]\[\mathcal{M} \hookrightarrow \widetilde{\mathfrak{F}} \quad (\simeq \mathfrak{F}_{\mathcal{M}})\]
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\[
\mathcal{M} \hookrightarrow \widetilde{\mathfrak{F}} \quad (\simeq \mathfrak{F}_{\mathcal{M}})
\]\[\mathrm{Fig}(\mathcal{M}) \longrightarrow \mathrm{Fig}(\mathcal{L})\]
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\[
\mathrm{Fig}(\mathcal{M}) \longrightarrow \mathrm{Fig}(\mathcal{L})
\]\[\mathcal{L} \longrightarrow \mathcal{M}\]
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\[
\mathcal{L} \longrightarrow \mathcal{M}
\]\[\mathfrak{F} \longrightarrow \mathrm{Fig}(\mathcal{L}), \qquad F \longmapsto \mathrm{multomb}(F)\]
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\[
\mathfrak{F} \longrightarrow \mathrm{Fig}(\mathcal{L}), \qquad F \longmapsto \mathrm{multomb}(F)
\]\[F = (X, \Sigma, \Phi)\]
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\[ F = (X, \Sigma, \Phi) \]
\[(X', \Sigma', \Phi') \leq (X, \Sigma, \Phi) \overset{\mathrm{df}}{\Longleftrightarrow} \begin{cases} \text{a) } X' \subset X,\ \text{sous-espace lin. par morceaux pour } \Sigma \\ \text{b) } \Sigma' \text{ induite par } \Sigma \\ \text{c) } \Phi' \leq \Phi \end{cases}\]
LaTeX source
\[
(X', \Sigma', \Phi') \leq (X, \Sigma, \Phi) \overset{\mathrm{df}}{\Longleftrightarrow} \begin{cases} \text{a) } X' \subset X,\ \text{sous-espace lin. par morceaux pour } \Sigma \\ \text{b) } \Sigma' \text{ induite par } \Sigma \\ \text{c) } \Phi' \leq \Phi \end{cases}
\]\[(X', \Sigma', \Phi') \ll (X, \Sigma, \Phi) \overset{\mathrm{df}}{\Longleftrightarrow} \begin{cases} \text{a) et b) comme dessus} \\ \text{c) } \Phi' \ll \Phi \end{cases}\]
LaTeX source
\[
(X', \Sigma', \Phi') \ll (X, \Sigma, \Phi) \overset{\mathrm{df}}{\Longleftrightarrow} \begin{cases} \text{a) et b) comme dessus} \\ \text{c) } \Phi' \ll \Phi \end{cases}
\]\[\begin{aligned}
(A', \sigma') \leq (A, \sigma) &\overset{\mathrm{df}}{\Longleftrightarrow} (A', \sigma') \text{ sous-simplexe (face) de } (A, \sigma) \\
(A', \sigma') \ll (A, \sigma) &\overset{\mathrm{df}}{\Longleftrightarrow} \mathfrak{L}_{\sigma'} \ll \mathfrak{L}_{\sigma} \\
(A, \sigma) \between (A', \sigma') &\Longleftrightarrow A \cap A' \text{ est vide ou une face commune de } (X, \sigma) \text{ et } (X', \sigma')
\end{aligned}\]
LaTeX source
\[
\begin{aligned}
(A', \sigma') \leq (A, \sigma) &\overset{\mathrm{df}}{\Longleftrightarrow} (A', \sigma') \text{ sous-simplexe (face) de } (A, \sigma) \\
(A', \sigma') \ll (A, \sigma) &\overset{\mathrm{df}}{\Longleftrightarrow} \mathfrak{L}_{\sigma'} \ll \mathfrak{L}_{\sigma} \\
(A, \sigma) \between (A', \sigma') &\Longleftrightarrow A \cap A' \text{ est vide ou une face commune de } (X, \sigma) \text{ et } (X', \sigma')
\end{aligned}
\]\[\mathcal{M} \longrightarrow \mathrm{Mulstrat}(\mathcal{L}) \subset \mathrm{Fig}(\mathcal{L})\]
LaTeX source
\[
\mathcal{M} \longrightarrow \mathrm{Mulstrat}(\mathcal{L}) \subset \mathrm{Fig}(\mathcal{L})
\]\[\begin{aligned}
\mathrm{multomb}(F) \leq \mathrm{multomb}(F') &\Longrightarrow F \leq F' \\
\mathrm{multomb}(F) \ll \mathrm{multomb}(F') &\Longrightarrow F \ll F'
\end{aligned}\]
LaTeX source
\[
\begin{aligned}
\mathrm{multomb}(F) \leq \mathrm{multomb}(F') &\Longrightarrow F \leq F' \\
\mathrm{multomb}(F) \ll \mathrm{multomb}(F') &\Longrightarrow F \ll F'
\end{aligned}
\]\[\mathfrak{F} \longrightarrow \mathrm{Fig}(\mathcal{L})\]
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\[
\mathfrak{F} \longrightarrow \mathrm{Fig}(\mathcal{L})
\]\[f_{\mathrm{mult}} : \mathcal{M} \longrightarrow \mathcal{M}'\]
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\[
f_{\mathrm{mult}} : \mathcal{M} \longrightarrow \mathcal{M}'
\]\[\mathrm{multomb} : \mathfrak{F} \longrightarrow \mathrm{Fig}(\mathcal{L})\]
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\[
\mathrm{multomb} : \mathfrak{F} \longrightarrow \mathrm{Fig}(\mathcal{L})
\]\[\mathcal{M}' = \mathfrak{F}' \cap \mathcal{M}\]
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\[
\mathcal{M}' = \mathfrak{F}' \cap \mathcal{M}
\]\[\mathcal{M}' = \mathcal{M} \cap \mathfrak{F}' .\]
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\[
\mathcal{M}' = \mathcal{M} \cap \mathfrak{F}' .
\]\[\forall F \in \mathfrak{F}, \quad F = \operatorname*{Sup}_{X \in \widetilde{F}} X \quad \text{où} \quad \widetilde{F} = \lbrace X \in \mathcal{M} \mid X \leq F \rbrace = \lbrace X \in \mathfrak{F},\ X \trianglelefteq F \rbrace\]
LaTeX source
\[
\forall F \in \mathfrak{F}, \quad F = \operatorname*{Sup}_{X \in \widetilde{F}} X \quad \text{où} \quad \widetilde{F} = \lbrace X \in \mathcal{M} \mid X \leq F \rbrace = \lbrace X \in \mathfrak{F},\ X \trianglelefteq F \rbrace
\]\[\begin{array}{ccc}
& \mathfrak{F}' \longmapsto \mathfrak{F}' \cap \mathcal{M} & \\
\text{Sousatspéc}(\mathfrak{F}) & \longrightarrow & \mathrm{Ferm}(\mathcal{M}) \\
& \longleftarrow & \\
\lbrace F \in \mathfrak{F} \mid \widetilde{F} \subset \mathcal{M}' \rbrace \overset{\mathrm{df}}{=} \mathrm{Env}(\mathcal{M}') & \longleftarrow & \mathcal{M}'
\end{array}\]
LaTeX source
\[
\begin{array}{ccc}
& \mathfrak{F}' \longmapsto \mathfrak{F}' \cap \mathcal{M} & \\
\text{Sousatspéc}(\mathfrak{F}) & \longrightarrow & \mathrm{Ferm}(\mathcal{M}) \\
& \longleftarrow & \\
\lbrace F \in \mathfrak{F} \mid \widetilde{F} \subset \mathcal{M}' \rbrace \overset{\mathrm{df}}{=} \mathrm{Env}(\mathcal{M}') & \longleftarrow & \mathcal{M}'
\end{array}
\]\[\mathrm{Multomb} : \mathfrak{F} \longrightarrow \mathrm{Fig}(\mathcal{M}),\]
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\[
\mathrm{Multomb} : \mathfrak{F} \longrightarrow \mathrm{Fig}(\mathcal{M}),
\]\[\mathfrak{F}'' \subset \mathfrak{F}' \subset \mathfrak{F}\]
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\[
\mathfrak{F}'' \subset \mathfrak{F}' \subset \mathfrak{F}
\]\[\begin{align*}
&(1.50) & \mathrm{Sousat}(\mathfrak{F}') &= \lbrace \mathfrak{F}'' \in \mathrm{Sousat}(\mathfrak{F}) \mid \mathfrak{F}'' \subset \mathfrak{F}' \rbrace
\end{align*}\]
LaTeX source
\begin{align*}
&(1.50) & \mathrm{Sousat}(\mathfrak{F}') &= \lbrace \mathfrak{F}'' \in \mathrm{Sousat}(\mathfrak{F}) \mid \mathfrak{F}'' \subset \mathfrak{F}' \rbrace
\end{align*}\[\begin{align*}
&(1.51) & \text{Sousatspéc}(\mathfrak{F}') &= \lbrace \mathfrak{F}'' \in \text{Sousatspéc}(\mathfrak{F}) \mid \mathfrak{F}'' \subset \mathfrak{F}' \rbrace .
\end{align*}\]
LaTeX source
\begin{align*}
&(1.51) & \text{Sousatspéc}(\mathfrak{F}') &= \lbrace \mathfrak{F}'' \in \text{Sousatspéc}(\mathfrak{F}) \mid \mathfrak{F}'' \subset \mathfrak{F}' \rbrace .
\end{align*}\[\begin{aligned}
\mathrm{omb}(X)^{\circ} &= \mathrm{omb}(X) \smallsetminus \mathrm{omb}(\partial X) \\
&= \mathrm{omb}(X) \smallsetminus \bigcup_{\substack{Y \trianglelefteq X \\ Y \neq X}} \mathrm{omb}(Y),
\end{aligned}\]
LaTeX source
\[
\begin{aligned}
\mathrm{omb}(X)^{\circ} &= \mathrm{omb}(X) \smallsetminus \mathrm{omb}(\partial X) \\
&= \mathrm{omb}(X) \smallsetminus \bigcup_{\substack{Y \trianglelefteq X \\ Y \neq X}} \mathrm{omb}(Y),
\end{aligned}
\]\[(1.52) \qquad \boxed{\ \mathrm{omb}(X)^{\circ} \neq \emptyset\ }\]
LaTeX source
\[
(1.52) \qquad \boxed{\ \mathrm{omb}(X)^{\circ} \neq \emptyset\ }
\]\[\mathrm{multomb}(F) = \lbrace \mathrm{omb}(X) \mid X \trianglelefteq F,\ \mathrm{omb}(X)^{\circ} \neq \emptyset \rbrace .\]
LaTeX source
\[
\mathrm{multomb}(F) = \lbrace \mathrm{omb}(X) \mid X \trianglelefteq F,\ \mathrm{omb}(X)^{\circ} \neq \emptyset \rbrace .
\]\[(1.53) \quad \left\lbrace
\begin{array}{c}
F' \longmapsto \mathrm{multomb}(F') \\
\mathrm{Ssfig}(F) \xrightarrow{\ \sim\ } \mathrm{Ssfig}(\mathrm{multomb}(F))
\end{array}
\right.\]
LaTeX source
\[
(1.53) \quad \left\lbrace
\begin{array}{c}
F' \longmapsto \mathrm{multomb}(F') \\
\mathrm{Ssfig}(F) \xrightarrow{\ \sim\ } \mathrm{Ssfig}(\mathrm{multomb}(F))
\end{array}
\right.
\]\[\mathcal{L} \cap \mathfrak{F}' \subset \mathcal{L}' \quad (\overset{\mathrm{df}}{=} \text{ens. des lieux de } \mathfrak{F}')\]
LaTeX source
\[
\mathcal{L} \cap \mathfrak{F}' \subset \mathcal{L}' \quad (\overset{\mathrm{df}}{=} \text{ens. des lieux de } \mathfrak{F}')
\]\[\mathcal{L} \cap \mathfrak{F}' = \mathcal{L}' .\]
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\[
\mathcal{L} \cap \mathfrak{F}' = \mathcal{L}' .
\]\[f : \mathfrak{F}' \longrightarrow \mathfrak{F}\]
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\[
f : \mathfrak{F}' \longrightarrow \mathfrak{F}
\]\[f_{\mathrm{str}} : \mathcal{M}' \longrightarrow \mathcal{M}\]
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\[
f_{\mathrm{str}} : \mathcal{M}' \longrightarrow \mathcal{M}
\]\[\begin{array}{ccc}
\mathrm{Ssfig}(F') & \longrightarrow & \mathrm{Ssfig}(F') \\
G' & \longmapsto & f(G')
\end{array}
\qquad \text{où } F = f(F')\]
LaTeX source
\[
\begin{array}{ccc}
\mathrm{Ssfig}(F') & \longrightarrow & \mathrm{Ssfig}(F') \\
G' & \longmapsto & f(G')
\end{array}
\qquad \text{où } F = f(F')
\]\[(1.56) \quad \left\lbrace
\begin{array}{c}
\mathfrak{P}_{\mathrm{f}}(\widetilde{F}') \longrightarrow \mathfrak{P}_{\mathrm{f}}(\widetilde{F}) \\
A \longmapsto \varphi^{-1}(A)
\end{array}
\right. ,\]
LaTeX source
\[
(1.56) \quad \left\lbrace
\begin{array}{c}
\mathfrak{P}_{\mathrm{f}}(\widetilde{F}') \longrightarrow \mathfrak{P}_{\mathrm{f}}(\widetilde{F}) \\
A \longmapsto \varphi^{-1}(A)
\end{array}
\right. ,
\]\[\mathrm{Hom}_{\mathrm{croiss}}(I, I') \longrightarrow \mathrm{Hom}_{\mathrm{cr}}(\mathfrak{P}_{\mathrm{f}}(I'), \mathfrak{P}_{\mathrm{f}}(I))\]
LaTeX source
\[
\mathrm{Hom}_{\mathrm{croiss}}(I, I') \longrightarrow \mathrm{Hom}_{\mathrm{cr}}(\mathfrak{P}_{\mathrm{f}}(I'), \mathfrak{P}_{\mathrm{f}}(I))
\]\[(*) \quad \mathfrak{P}_{\mathrm{f}}(I') \longrightarrow \mathfrak{P}_{\mathrm{f}}(I)\]
LaTeX source
\[
(*) \quad \mathfrak{P}_{\mathrm{f}}(I') \longrightarrow \mathfrak{P}_{\mathrm{f}}(I)
\]\[\mathrm{Top}(I) \longrightarrow \mathrm{Top}(I')\]
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\[
\mathrm{Top}(I) \longrightarrow \mathrm{Top}(I')
\]\[(\mathcal{M}, \leq, \ll \overset{\mathrm{df}}{=} \leq, \mathfrak{F}).\]
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\[
(\mathcal{M}, \leq, \ll \overset{\mathrm{df}}{=} \leq, \mathfrak{F}).
\]\[(\mathcal{M}', \leq, \ll \overset{\mathrm{df}}{=} \leq, \mathfrak{F}'),\]
LaTeX source
\[
(\mathcal{M}', \leq, \ll \overset{\mathrm{df}}{=} \leq, \mathfrak{F}'),
\]\[f : \mathfrak{F}' \longrightarrow \mathfrak{F}\]
LaTeX source
\[
f : \mathfrak{F}' \longrightarrow \mathfrak{F}
\]\[\begin{array}{rcl}
f^{*} : \mathrm{Fig}(\mathcal{L}') & \longrightarrow & \mathrm{Fig}(\mathcal{L}) \\
\Phi' & \longmapsto & \lbrace f^{-1}(A') \mid A' \in \Phi',\ f^{-1}(A') \neq \emptyset \rbrace
\end{array}\]
LaTeX source
\[
\begin{array}{rcl}
f^{*} : \mathrm{Fig}(\mathcal{L}') & \longrightarrow & \mathrm{Fig}(\mathcal{L}) \\
\Phi' & \longmapsto & \lbrace f^{-1}(A') \mid A' \in \Phi',\ f^{-1}(A') \neq \emptyset \rbrace
\end{array}
\]\[\mathrm{Fig}(\mathcal{M}) \longrightarrow \mathrm{Fig}(\mathcal{L})\]
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\[
\mathrm{Fig}(\mathcal{M}) \longrightarrow \mathrm{Fig}(\mathcal{L})
\]\[\mathfrak{F} \longrightarrow \mathrm{Fig}(\mathcal{L})\]
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\[
\mathfrak{F} \longrightarrow \mathrm{Fig}(\mathcal{L})
\]\[(*) \quad \mathrm{Sousfig}(F') \longrightarrow \mathrm{Sousfig}(F), \qquad G' \longmapsto f(G')\]
LaTeX source
\[
(*) \quad \mathrm{Sousfig}(F') \longrightarrow \mathrm{Sousfig}(F), \qquad G' \longmapsto f(G')
\]\[f(G') \ll f(F') \Longrightarrow G' \ll F'\]
LaTeX source
\[
f(G') \ll f(F') \Longrightarrow G' \ll F'
\]\[f(G') \leq f(F') \Longrightarrow G' \leq F'\]
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\[ f(G') \leq f(F') \Longrightarrow G' \leq F' \]
\[\mathfrak{F} \longrightarrow \mathrm{Fig}(\mathcal{L})\]
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\[
\mathfrak{F} \longrightarrow \mathrm{Fig}(\mathcal{L})
\]\[\mathrm{multomb}(F) \ll \mathrm{multomb}(G) \Longrightarrow F \ll G .\]
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\[
\mathrm{multomb}(F) \ll \mathrm{multomb}(G) \Longrightarrow F \ll G .
\]\[x \longmapsto \lbrace \lbrace x \rbrace \rbrace = \Phi_x\]
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\[ x \longmapsto \lbrace \lbrace x \rbrace \rbrace = \Phi_x \]
\[(\mathcal{L}, \mathcal{M}),\]
LaTeX source
\[
(\mathcal{L}, \mathcal{M}),
\]\[F \parallel G \Longleftrightarrow \forall X \in \widetilde{F},\ Y \in \widetilde{G},\quad X \parallel Y .\]
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\[
F \parallel G \Longleftrightarrow \forall X \in \widetilde{F},\ Y \in \widetilde{G},\quad X \parallel Y .
\]\[A' \subsetneq A, \qquad A \cap B = \emptyset .\]
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\[ A' \subsetneq A, \qquad A \cap B = \emptyset . \]
\[X = \lbrace A \rbrace, \quad Y = \lbrace B \rbrace, \quad X' = \lbrace A' \rbrace, \quad \text{et soit } Y' = Y,\]
LaTeX source
\[
X = \lbrace A \rbrace, \quad Y = \lbrace B \rbrace, \quad X' = \lbrace A' \rbrace, \quad \text{et soit } Y' = Y,
\]\[X' \ll X, \qquad Y' = Y \text{ donc } Y' \ll Y .\]
LaTeX source
\[
X' \ll X, \qquad Y' = Y \text{ donc } Y' \ll Y .
\]\[X \underset{\mathcal{M}}{\between} Y, \quad \text{et} \quad X' \underset{\mathcal{M}}{\overline{\between}} Y\]
LaTeX source
\[
X \underset{\mathcal{M}}{\between} Y, \quad \text{et} \quad X' \underset{\mathcal{M}}{\overline{\between}} Y
\]\[X \underset{\mathfrak{F}}{\parallel} Y, \quad \text{mais} \quad X' \underset{\mathfrak{F}}{\overline{\parallel}} Y, \quad \text{ok.}\]
LaTeX source
\[
X \underset{\mathfrak{F}}{\parallel} Y, \quad \text{mais} \quad X' \underset{\mathfrak{F}}{\overline{\parallel}} Y, \quad \text{ok.}
\]\[\mathrm{omb}(X) \cap \mathrm{omb}(Y) = \mathrm{omb}(F) \,]\]
LaTeX source
\[
\mathrm{omb}(X) \cap \mathrm{omb}(Y) = \mathrm{omb}(F) \,]
\]\[\begin{equation}
x \parallel y \Longleftrightarrow x \between y \text{ et } x \neq y \tag{1.59}
\end{equation}\]
LaTeX source
\begin{equation}
x \parallel y \Longleftrightarrow x \between y \text{ et } x \neq y \tag{1.59}
\end{equation}\[X \between Y \Longleftrightarrow
\begin{cases}
\forall\, x \in \mathrm{omb}\, X \smallsetminus \mathrm{omb}\, F \text{ et } y \in \mathrm{omb}\, Y \smallsetminus \mathrm{omb}\, F, \text{ on a } x \parallel y \\
F \text{ est « élémentaire » (i.e. } F \in \mathcal{M} \text{)}
\end{cases}\]
LaTeX source
\[
X \between Y \Longleftrightarrow
\begin{cases}
\forall\, x \in \mathrm{omb}\, X \smallsetminus \mathrm{omb}\, F \text{ et } y \in \mathrm{omb}\, Y \smallsetminus \mathrm{omb}\, F, \text{ on a } x \parallel y \\
F \text{ est « élémentaire » (i.e. } F \in \mathcal{M} \text{)}
\end{cases}
\]\[G' = \mathrm{Inf}^{\ll}(F', G)\]
LaTeX source
\[
G' = \mathrm{Inf}^{\ll}(F', G)
\]\[\begin{equation}
\widetilde{G'} = \lbrace X \in \widetilde{F'} \mid X \ll G \rbrace \tag{1.60}
\end{equation}\]
LaTeX source
\begin{equation}
\widetilde{G'} = \lbrace X \in \widetilde{F'} \mid X \ll G \rbrace \tag{1.60}
\end{equation}\[\mathfrak{X} \ll F',\ \mathfrak{X} \ll G \Longrightarrow \mathfrak{X} \ll G'\]
LaTeX source
\[
\mathfrak{X} \ll F',\ \mathfrak{X} \ll G \Longrightarrow \mathfrak{X} \ll G'
\]\[F' \ll F, \quad X' \leq F', \quad G \leq F, \quad Z \overset{\circ}{\ll} X', \quad Z \ll G,\]
LaTeX source
\[
F' \ll F, \quad X' \leq F', \quad G \leq F, \quad Z \overset{\circ}{\ll} X', \quad Z \ll G,
\]\[Z \overset{\circ}{\ll} X\]
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\[
Z \overset{\circ}{\ll} X
\]\[\begin{equation}
\mathrm{Sousfig}(F) \longrightarrow \mathrm{Sousfig}(F') \tag{1.61}
\end{equation}\]
LaTeX source
\begin{equation}
\mathrm{Sousfig}(F) \longrightarrow \mathrm{Sousfig}(F') \tag{1.61}
\end{equation}\[\varphi : \widetilde{F'} \longrightarrow \widetilde{F}\]
LaTeX source
\[
\varphi : \widetilde{F'} \longrightarrow \widetilde{F}
\]\[\varphi^{*} : \mathfrak{P}_{\mathrm{f}}(\widetilde{F}) \longrightarrow \mathfrak{P}_{\mathrm{f}}(\widetilde{F'}), \qquad A \longmapsto \varphi^{-1}(A) \,)\]
LaTeX source
\[
\varphi^{*} : \mathfrak{P}_{\mathrm{f}}(\widetilde{F}) \longrightarrow \mathfrak{P}_{\mathrm{f}}(\widetilde{F'}), \qquad A \longmapsto \varphi^{-1}(A) \,)
\]\[F'_Y = F'_X | Y\]
LaTeX source
\[ F'_Y = F'_X | Y \]
\[\operatorname*{Sup}_{X \in \widetilde{F}} F'_X = F'\]
LaTeX source
\[
\operatorname*{Sup}_{X \in \widetilde{F}} F'_X = F'
\]\[X' | T = Y' | T\]
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\[ X' | T = Y' | T \]
\[\widetilde{F'} = \bigcup_{X} \widetilde{F'_X}\]
LaTeX source
\[
\widetilde{F'} = \bigcup_{X} \widetilde{F'_X}
\]\[F'_X = \operatorname*{Sup}_{Z \in \widetilde{F}} (F'_Z | X)\]
LaTeX source
\[
F'_X = \operatorname*{Sup}_{Z \in \widetilde{F}} (F'_Z | X)
\]\[\widetilde{F'} = \bigcup \widetilde{F'_X} \longrightarrow \widetilde{F} \supset \widetilde{X}\]
LaTeX source
\[
\widetilde{F'} = \bigcup \widetilde{F'_X} \longrightarrow \widetilde{F} \supset \widetilde{X}
\]\[H' \overset{\mathrm{df}}{=} F' \amalg G' \ll H = F \amalg G .\]
LaTeX source
\[
H' \overset{\mathrm{df}}{=} F' \amalg G' \ll H = F \amalg G .
\]\[\mathrm{Raff}(F) \times \mathrm{Raff}(G) \xrightarrow{\ \sim\ } \mathrm{Raff}(F \amalg G), \qquad (F', G') \longmapsto H' = F' \amalg G'\]
LaTeX source
\[
\mathrm{Raff}(F) \times \mathrm{Raff}(G) \xrightarrow{\ \sim\ } \mathrm{Raff}(F \amalg G), \qquad (F', G') \longmapsto H' = F' \amalg G'
\]\[H' \longmapsto (H'|F, H'|G) .\]
LaTeX source
\[
H' \longmapsto (H'|F, H'|G) .
\]\[F \parallel G, \quad F' \ll F,\ G' \ll G \Longrightarrow F' \parallel G' !\]
LaTeX source
\[ F \parallel G, \quad F' \ll F,\ G' \ll G \Longrightarrow F' \parallel G' ! \]
\[K' \leq F' \ll F \quad \text{et} \quad K' \leq G' \ll G, \quad \text{donc} \quad K'' \ll F, G,\]
LaTeX source
\[
K' \leq F' \ll F \quad \text{et} \quad K' \leq G' \ll G, \quad \text{donc} \quad K'' \ll F, G,
\]\[K' = \mathrm{Inf}^{\ll}(F', G') = \mathrm{Inf}^{\ll}(F', G', K) = \mathrm{Inf}^{\ll}\bigl(\underbrace{\mathrm{Inf}(F', K)}_{F'|K}, \underbrace{\mathrm{Inf}(G', K)}_{G'|K}\bigr)\]
LaTeX source
\[
K' = \mathrm{Inf}^{\ll}(F', G') = \mathrm{Inf}^{\ll}(F', G', K) = \mathrm{Inf}^{\ll}\bigl(\underbrace{\mathrm{Inf}(F', K)}_{F'|K}, \underbrace{\mathrm{Inf}(G', K)}_{G'|K}\bigr)
\]\[K' = \mathrm{Inf}^{\ll}(F', G') = \mathrm{Inf}^{\ll}(F', G', K) = \mathrm{Inf}^{\ll}\bigl(\mathrm{Inf}^{\ll}(F', K), \mathrm{Inf}^{\ll}(G', K)\bigr)\]
LaTeX source
\[
K' = \mathrm{Inf}^{\ll}(F', G') = \mathrm{Inf}^{\ll}(F', G', K) = \mathrm{Inf}^{\ll}\bigl(\mathrm{Inf}^{\ll}(F', K), \mathrm{Inf}^{\ll}(G', K)\bigr)
\]\[K' = \mathrm{Inf}(\underbrace{F'|K, G'|K}_{\text{deux sous-figures de } H'}) = (\underbrace{F'|K}_{F'_K}) \cap (\underbrace{G'|K}_{G'_K}) .\]
LaTeX source
\[
K' = \mathrm{Inf}(\underbrace{F'|K, G'|K}_{\text{deux sous-figures de } H'}) = (\underbrace{F'|K}_{F'_K}) \cap (\underbrace{G'|K}_{G'_K}) .
\]\[H'|F = F' \vee G'_K, \qquad H'|G = G' \vee F'_K\]
LaTeX source
\[ H'|F = F' \vee G'_K, \qquad H'|G = G' \vee F'_K \]
\[H'_F = F' \vee H'_K, \qquad = G' \vee H'_K\]
LaTeX source
\[ H'_F = F' \vee H'_K, \qquad = G' \vee H'_K \]
\[F' \ll F \quad \text{et} \quad G' \ll G\]
LaTeX source
\[
F' \ll F \quad \text{et} \quad G' \ll G
\]\[H'_K = F'_K \vee G'_K \qquad \text{on aura } H'_K \ll K \text{ par At 4}\]
LaTeX source
\[
H'_K = F'_K \vee G'_K \qquad \text{on aura } H'_K \ll K \text{ par At 4}
\]\[K' = \underbrace{F'_K \cap G'_K}\]
LaTeX source
\[
K' = \underbrace{F'_K \cap G'_K}
\]\[X \supseteq Y, \quad X' \overset{\circ}{\ll} X, \quad Y' \overset{\circ}{\ll} Y, \quad X' \supseteq X'_Y = X'|Y .\]
LaTeX source
\[
X \supseteq Y, \quad X' \overset{\circ}{\ll} X, \quad Y' \overset{\circ}{\ll} Y, \quad X' \supseteq X'_Y = X'|Y .
\]\[X'_K \leq X', \quad X'_K \ll K \quad \text{et} \quad Y'_K \leq Y', \quad Y'_K \ll K .\]
LaTeX source
\[
X'_K \leq X', \quad X'_K \ll K \quad \text{et} \quad Y'_K \leq Y', \quad Y'_K \ll K .
\]\[K = X \cap Y \quad \Bigl( = \mathrm{Inf}^{\leq}(X, Y) = \operatorname*{Sup}_{Z \in \widetilde{X} \cap \widetilde{Y}} Z \Bigr),\]
LaTeX source
\[
K = X \cap Y \quad \Bigl( = \mathrm{Inf}^{\leq}(X, Y) = \operatorname*{Sup}_{Z \in \widetilde{X} \cap \widetilde{Y}} Z \Bigr),
\]\[\begin{array}{lll}
\text{At } 9_{\Sigma} \Longrightarrow \text{At } 9 & & \text{At } 11 \Longleftrightarrow \text{At } 11 \text{ pour } F = \operatorname{Sup} F_i \text{ quelc.} \\
\Downarrow & & \\
\text{At } 12 \Longrightarrow \text{At } 10 & & \text{At } 11' \Longleftrightarrow \text{At } 11 \text{ pour } F = \operatorname{Sup} F_i \text{ fini} \\
\text{At } 12 \Longrightarrow \text{At } 11' \text{ pour } \widetilde{F} \text{ fini} & &
\end{array}\]
LaTeX source
\[
\begin{array}{lll}
\text{At } 9_{\Sigma} \Longrightarrow \text{At } 9 & & \text{At } 11 \Longleftrightarrow \text{At } 11 \text{ pour } F = \operatorname{Sup} F_i \text{ quelc.} \\
\Downarrow & & \\
\text{At } 12 \Longrightarrow \text{At } 10 & & \text{At } 11' \Longleftrightarrow \text{At } 11 \text{ pour } F = \operatorname{Sup} F_i \text{ fini} \\
\text{At } 12 \Longrightarrow \text{At } 11' \text{ pour } \widetilde{F} \text{ fini} & &
\end{array}
\]\[F' = \operatorname{Sup} F'_i\]
LaTeX source
\[
F' = \operatorname{Sup} F'_i
\]\[\text{At } 9_{\Sigma}, \quad \text{At } 10, \quad \text{At } 11 \text{ bis}\]
LaTeX source
\[
\text{At } 9_{\Sigma}, \quad \text{At } 10, \quad \text{At } 11 \text{ bis}
\]\[F' \ll F, \qquad F'_X \leq F', \qquad X \leq F, \qquad F'_X \ll X, \qquad x \mathrel{\overset{\circ}{\ll}} X, \quad y\]
LaTeX source
\[
F' \ll F, \qquad F'_X \leq F', \qquad X \leq F, \qquad F'_X \ll X, \qquad x \mathrel{\overset{\circ}{\ll}} X, \quad y
\]\[F'_X \ll X, \qquad x \mathrel{\overset{\circ}{\ll}} X, \qquad y \qquad (\text{i.e. on } F \in \mathcal{M},\ G \in \mathcal{L})\]
LaTeX source
\[
F'_X \ll X, \qquad x \mathrel{\overset{\circ}{\ll}} X, \qquad y \qquad (\text{i.e. on } F \in \mathcal{M},\ G \in \mathcal{L})
\]\[F' \ll F, \qquad F'_G \leq F', \qquad G \leq F, \qquad F'_G \ll G, \qquad H \ll G\]
LaTeX source
\[ F' \ll F, \qquad F'_G \leq F', \qquad G \leq F, \qquad F'_G \ll G, \qquad H \ll G \]
\[x' \in \mathrm{omb}\, F',\ x \in \mathrm{omb}\, H \Longrightarrow x \parallel x' .\]
LaTeX source
\[
x' \in \mathrm{omb}\, F',\ x \in \mathrm{omb}\, H \Longrightarrow x \parallel x' .
\]\[x' \ll F' \ll F, \qquad x'_G \ll F'_G \ll G, \qquad x \ll G\]
LaTeX source
\[ x' \ll F' \ll F, \qquad x'_G \ll F'_G \ll G, \qquad x \ll G \]
\[F' \leq \overline{F'} \ll F\]
LaTeX source
\[
F' \leq \overline{F'} \ll F
\]\[F' \vee x = \overline{F'}\]
LaTeX source
\[
F' \vee x = \overline{F'}
\]\[F' \preccurlyeq F .\]
LaTeX source
\[ F' \preccurlyeq F . \]
\[F'' \preccurlyeq F' \preccurlyeq F \Longrightarrow F'' \preccurlyeq F .\]
LaTeX source
\[ F'' \preccurlyeq F' \preccurlyeq F \Longrightarrow F'' \preccurlyeq F . \]
\[F' \ll F, \qquad F'_G \leq F', \qquad G \leq F, \qquad F'_G \ll G .\]
LaTeX source
\[ F' \ll F, \qquad F'_G \leq F', \qquad G \leq F, \qquad F'_G \ll G . \]
\[F = X \in \mathcal{M} .\]
LaTeX source
\[
F = X \in \mathcal{M} .
\]\[\widetilde{G} \subset \widetilde{F}, \qquad \widetilde{G'} \longrightarrow \widetilde{G}\]
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\[
\widetilde{G} \subset \widetilde{F}, \qquad \widetilde{G'} \longrightarrow \widetilde{G}
\]\[\widetilde{F'} \smallsetminus \widetilde{G'_{+}} \xrightarrow{\ \sim\ } \widetilde{F} \smallsetminus \widetilde{G},\]
LaTeX source
\[
\widetilde{F'} \smallsetminus \widetilde{G'_{+}} \xrightarrow{\ \sim\ } \widetilde{F} \smallsetminus \widetilde{G},
\]\[\mathfrak{F} \subset \mathrm{Fig}(\mathcal{L})\]
LaTeX source
\[
\mathfrak{F} \subset \mathrm{Fig}(\mathcal{L})
\]\[\mathcal{M} = \lbrace \lbrace \lbrace x \rbrace \rbrace \mid x \in \mathcal{L} \rbrace ,\]
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\[
\mathcal{M} = \lbrace \lbrace \lbrace x \rbrace \rbrace \mid x \in \mathcal{L} \rbrace ,
\]\[\mathcal{L} \cap \mathfrak{F}' \subset \mathcal{L}' .\]
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\[
\mathcal{L} \cap \mathfrak{F}' \subset \mathcal{L}' .
\]\[F \in \mathfrak{F}',\ x \ll F \ (x \in \mathcal{L}) \Longrightarrow x \in \mathfrak{F}' .\]
LaTeX source
\[
F \in \mathfrak{F}',\ x \ll F \ (x \in \mathcal{L}) \Longrightarrow x \in \mathfrak{F}' .
\]\[\mathcal{L}' = \mathcal{L} \cap \mathfrak{F}' .\]
LaTeX source
\[
\mathcal{L}' = \mathcal{L} \cap \mathfrak{F}' .
\]