\[
  |\mathfrak{L}| = \bigcup_{A \in \mathfrak{L}} A
\]

\[
  \partial A = \bigcup_{\substack{B \in \mathfrak{L} \\ B \subset A}} B , \qquad A^{\circ} = A \smallsetminus \partial A .
\]

\[
  \mathfrak{L}_x = \{ A \in \mathfrak{L} \mid x \in A \}
\]

\[
  \mathfrak{L}' = \{ A \in \mathfrak{L} \mid A^{\circ} \neq \emptyset \} )
\]

\[
  A_1 \supset A_2 \supset \cdots
\]

\[
  \struck{\Bigl\{ \bigcup_{A \in \mathfrak{L}'} A^{\circ} = |\mathfrak{L}| , \quad \forall A \in \mathfrak{L}, \text{ on a } A = \bigcup_{B \in \mathfrak{L} \text{ tq. } B^{\circ} \subset A} B^{\circ}}
\]

\[
  \mathfrak{L} = \bigl\{ \overline{\varphi^{-1}(i)} = \varphi^{-1}(\overline{\{i\}} = I_{\leq i}) \bigr\}_{i \in I}
\]

\[
  A_J = \bigcup_{i \in J} A_i = \bigcup_{A \in J} A = \underbrace{\coprod_{i \in J} A_i^{\circ}}_{\text{réunion disjointe de parties non vides}}
\]

\[
  (1) \qquad A_J = \bigcup_{i \in J} A_i^{\circ} = \varphi^{-1}(J)
\]

\[
  A_J = \varphi^{-1}(J) = \bigcup_{i \in J} A_i^{\circ}
\]

\[
  (*) \qquad i \leq j \leq k, \quad i, k \in J \Longrightarrow j \in J
\]

\[
  J \longmapsto A_J = \varphi^{-1}(J) = \bigcup_{i \in J} A_i^{\circ} .
\]

\[
  \partial_{\Psi} A = \partial_{\mathfrak{L}} A \quad \text{donc} \quad A^{\circ (\Psi)} = A^{\circ (\mathfrak{L})} ,
\]

\[
  |\Psi| = A_{\Psi}
\]

\[
  A \in \Psi, \quad B \in \mathfrak{L}, \quad B \subset A \quad \text{prouvons } B \in \Psi
\]

\[
  |\mathfrak{L}| \cap |\Psi| = |\mathfrak{L} \cap \Psi|
\]

\[
  \bigcup_{\substack{B \in \mathfrak{L} \cup \Psi \\ B \subsetneq A}} B \neq A .
\]

\[
  B \subset A \Longrightarrow B \in \mathfrak{L} .
\]

\[
  |\mathfrak{L}| \subset |\Psi| \ \text{ssi}\ \mathfrak{L} \subset \Psi , \qquad |\mathfrak{L}| = |\Psi| \ \text{ssi}\ \mathfrak{L} = \Psi .
\]

\[
  \mathfrak{L} \leq \Psi \overset{\text{déf}}{\Longleftrightarrow} \mathfrak{L} \text{ est une sous-famille admissible de } \Psi .
\]

\[
  \mathfrak{L}_X = \{ Y \in \mathfrak{L} \mid Y \leq X \} , \quad \text{pour } X \in \mathfrak{L}
\]

\[
  \mathfrak{L} = \{A\}
\]

\[
  \mathfrak{X}_A = \{A\} \leq F
\]

\[
  \bigl\{ \{A\}, \{\{x\}\}, A \cup \{x\} \bigr\}
\]

\[
  \bigl\{ \{\{x\}\}, \{L \smallsetminus \{x\}\} \bigr\}
\]

\[
  \mathcal{M} = \{\{L\}\}
\]

\[
  |X_A| = A
\]

\[
  Z = \bigl\{ B, B \cup \{x\}, \{x\} \bigm| B \in X_A \bigr\} ,
\]

\[
  F \boxtimes G = \bigl\{ A \cup B \bigm| \underbrace{A \in F, B \in G}_{A \text{ et } B \text{ pouvant être } \emptyset} \bigr\} ,
\]

\[
  (A \cup B)^{\circ} = A^{\circ} \cup B^{\circ} , \qquad \partial(A \cup B) = (\partial A \cup B) \cup (A \cup \partial B)
\]

\[
  |F| \cap |G| = \emptyset .
\]

\[
  L' = L \smallsetminus \bigcup_i |G_i| .
\]

\[
  \mathfrak{F}(L') \subset \mathfrak{F}(L)
\]

\[
  \mathcal{U} = \operatorname{int}(\complement \mathcal{V}) = \operatorname{ext} \mathcal{V} , \qquad \mathcal{V} = \operatorname{int}(\complement \mathcal{U}) = \operatorname{ext} \mathcal{U} ,
\]

\[
  X \smallsetminus (\mathcal{U} \cup \mathcal{V}) = \partial \mathcal{U} \cap \partial \mathcal{V} ,
\]

\[
  0 \leq i \leq +\infty ,
\]

\[
  A \longmapsto X_A
\]

\[
  \mathfrak{L} \subset \mathfrak{P}_{\text{fermés}}(L)
\]

\[
  A^{\circ} \subset B^{\circ} , \quad A'^{\circ} \subset B'^{\circ} , \quad A' \subset A \Longrightarrow B' \subset B .
\]

\[
  A = \bigcup_{\substack{A' \in F \\ A' \subset A}} A'^{\circ}
\]

\[
  A \cap B = \bigcup_{\substack{C \in F \\ \text{t.q. } C^{\circ} \subset A \cap B}} C^{\circ} .
\]

\[
  C \in F, \quad C^{\circ} \subset A \cap B \Longrightarrow C \subset A \cap B .
\]

\[
  A, A' \in F, \quad B, B' \in G, \quad A^{\circ} \subset B^{\circ}, \quad A'^{\circ} \subset B'^{\circ}, \quad A' \subset A^{*} \Longrightarrow B' \subset B^{*}
\]

\[
  \begin{cases}
    H \neq \emptyset \\
    H \text{ raffine } G \\
    H \text{ disjointe de } F
  \end{cases}
\]

\[
  p : S \to I, \qquad q : T \to J
\]

\[
  J' \mapsto \varphi^{-1}(J')
\]

\[
    F \leq G' \preccurlyeq G
  \]

\[
  G' = \bigl\{A \in \mathfrak{P}(\mathcal{L}) \bigm| A \in F \text{ ou } A = \{x\}, \text{ avec } x \in |G| \smallsetminus |F|\bigr\}
  = F \amalg \coprod_{x \in |G| \smallsetminus |F|} \{\{x\}\}
\]

\[
  F \leq G \text{ et } F \preccurlyeq G \Longrightarrow F = G .
\]

\[
  H \ll G, \quad \underset{\text{i.e. } H \text{ raffine } G \text{ et } H \cap G = \emptyset}{H \text{ disjointe de } G} \Longrightarrow H = \emptyset_{F}
\]

\[
  K \struck{I} = \{(A,B) \in F \times G \mid A^{\circ} \cap B^{\circ} \neq \emptyset\}
\]

\[
  |F| \cap |G| = \coprod_{(A,B) \in K} A^{\circ} \cap B^{\circ}
\]

\[
  k = (A,B), \qquad C_{k} = A \cap B
\]

\[
  K = \operatorname{Im}\bigl(S \cap T \xrightarrow{(\varphi,\varphi')} I \times J\bigr) = \{(i,i') \in I \times I' \mid A_{i}^{\circ} \cap A'^{\circ}_{i'} \neq \emptyset\}
\]

\[
  \boxed{A_{j} = A_{i} \cap A'_{i'}} \qquad \text{où on pose } A_{i} = \varphi^{-1}(I_{\leq i}), \quad A'_{i'} = \varphi'^{-1}(I'_{\leq i'})
\]

\[
  \Bigl|\bigwedge_{i \in I} F_{i}\Bigr| = \bigcap_{i \in I} |F_{i}| .
\]

\[
  \underset{F'}{A} \cap \underset{G}{B} \in F'.G \quad \text{i.e. } A^{\circ} \cap B^{\circ} \neq \emptyset, \qquad
  \underset{F}{A'} \cap \underset{G}{B'} \in F.G \quad \text{i.e. } A'^{\circ} \cap B'^{\circ} \neq \emptyset,
\]

\[
  G \leq F, \quad G' \preccurlyeq G \qquad \exists \text{ subdivision } F' \text{ de } F \text{ qui induise } G'.
\]

\[
  |F'| = \underbrace{|G'|}_{|G|} \cup \underbrace{|F \smallsetminus G|}_{\supset |F| \smallsetminus |G|} \supset |F|
\]

\[
  |F'| = |F| .
\]

\[
    F' \smallsetminus G' \xrightarrow{\ \sim\ } F \smallsetminus G .
  \]

\[
  \text{d)} \Longrightarrow \text{b)} \Longrightarrow \text{c)} \Longrightarrow \text{a)} \Longrightarrow \text{d)}
\]

\[
  F \smallsetminus G = F' \smallsetminus G' .
\]

\[
  S = |F| = |F'| = |F''|
\]

\[
  |G'| = |G| = T
\]

\[
  S \xrightarrow{\ g\ } I' \xrightarrow{\ h\ } I \qquad g \text{ surj}, \quad h \text{ croiss. surj.}
\]

\[
  K = J' \amalg I \smallsetminus J \quad \text{comme ens. \ill{}}
\]

\[
  \begin{cases}
    \alpha \leq \beta \text{ ssi } h_{J}(\alpha) \leq \beta \text{ dans } I \\
    \struck{\ill{}}
  \end{cases}
\]

\[
  K \smallsetminus J' \xrightarrow{\ \sim\ } I \smallsetminus J ,
\]

\[
  (S',T') \ll (S,T)
\]

\[
  (S',T') \preccurlyeq (S,T)
\]

\[
  G \subset F, \qquad G' \preccurlyeq G
\]

\[
  \struck{\ill{}}\ T \hookrightarrow S, \qquad \mathcal{T}_{0} \text{ sur } T, \quad \mathcal{T} \text{ sur } S, \qquad \mathcal{T}'_{0} \leq \mathcal{T}_{0}
\]

\[
  \underset{\mathcal{T}'_{0}}{T} \xrightarrow{\ i\ } \underset{\mathcal{T}'\,?}{S} \xrightarrow{\ \mathrm{id}\ } \underset{\mathcal{T}}{S}
\]

\[
  \begin{cases}
    A \subset S \text{ fermés pour } \mathcal{T} \\
    B \subset T \text{ fermés pour } \mathcal{T}'_{0},
  \end{cases}
\]

\[
  (A \cup B) \cap T = (A \cap T) \cup B
\]

\[
  \bigcap_{i \in I} (A_{i} \cup B_{i}) = \bigcup_{J \subset I} \Bigl(\underbrace{\bigcap_{i \in J} A_{i}}_{A}\Bigr) \cap \Bigl(\underbrace{\bigcap_{j \in I \smallsetminus J} B_{j}}_{B}\Bigr)
\]

\[
  C = \underbrace{(C \cap T)}_{\text{fermé de } (T, \mathcal{T}'_{0}),\ \text{soit } B} \cup \underbrace{\overline{(C \cap (S \smallsetminus T))}^{(\mathcal{T}')}}_{A}
\]

\[
  (B, A_{\mathcal{U}}) \qquad
  \begin{array}{l}
    B \subset T \text{ fermé pour } \tau'_0 \\
    A_{\mathcal{U}} \text{ fermé de } \mathcal{U} = S \smallsetminus T \text{ pour } \tau|(S \smallsetminus T)
  \end{array}
\]

\[
  \overline{A_{\mathcal{U}}} \cap T \subset B \qquad (\text{adh. prise pour } \tau)
\]

\[
  \begin{cases}
    \text{ou bien } x, y \in T, & x \leq_{\uncertain{\tau'_0}} y \\
    \text{ou bien } x, y \in \mathcal{U} = S \smallsetminus T, & x \leq_{\tau} y \\
    \text{ou bien } x \in T,\ y \in \mathcal{U}, & x \leq_{\tau} y
  \end{cases}
\]

\[
  X = B \cup \overline{X \cap \mathcal{U}}^{(\tau)}
\]

\[
  X = B \cup \overline{X \cap \mathcal{U}}^{(\tau)}
\]

\[
  x \leq_{\tau'} y \iff \overline{x}^{(\tau')} \subset \overline{y}^{(\tau')} \iff x \in \overline{y}^{(\tau')} .
\]

\[
  X \supset \overline{X \cap \mathcal{U}}^{(\tau)} ,
\]

\[
  X \supset \overline{A}^{(\tau)} , \qquad \struck{\ill{}} \text{ en particulier}
\]

\[
  \overline{A}^{(\tau')} \supset \overline{A}^{(\tau)} , \qquad \text{OK.}
\]

\[
  S \xrightarrow{\ \mathrm{id}\ } S, \qquad T \xrightarrow{\ \mathrm{inc.}\ } S
\]

\[
  \overline{X}^{(\tau')} = \overline{X \cap \mathcal{U}}^{(\tau)} \cup \overline{X \cap T}^{(\tau'_0)}
\]

\[
  X = \uncertain{\text{second membre}} = \overline{X \cap \mathcal{U}}^{(\tau)} \cup (X \cap T)
\]

\[
  \varphi^* : \mathcal{J}' \to \mathcal{J} \text{ surjectif.}
\]

\[
  \varphi^{-1}(J_{\leq x})
\]

\[
  \overline{X_{\mathcal{U}}}^{(\tau)} \subset \overline{X_{\mathcal{U}}}^{(\tau'')} \qquad \left(\subset \overline{X}^{(\tau'')} = X\right)
\]

\[
  \overline{A}^{(\tau'')} = \overline{A}^{(\tau)} , \quad \text{i.e.} \quad \overline{A}^{(\tau'')} \text{ est } \tau\text{-fermé}
\]

\[
  \overline{A}^{(\tau'')} \cap T \text{ est } \tau \text{ fermé} \qquad ??
\]

\[
  \overline{A}^{(\tau')} = \overline{A \cap \mathcal{U}}^{(\tau)} \cap \overline{(A \cap T)}^{(\tau'_0)}
\]