\[ u^{*-1}(B) \cap A \in \mathcal{M}_m \]

\[ A \times B \in \mathcal{M}_{m+n} \]

\[ A, B \in \mathfrak{S}_n \Rightarrow A \cap B \in \mathfrak{S}_n \]

\[ (*) \qquad \mathcal{M}_1 \neq \emptyset . \]

\[ (**) \qquad \emptyset \in \mathcal{M}_1 \quad (\text{d'où } \emptyset \in
\mathcal{M}_n \ \forall n \in \mathbb{N}) \]

\[ \mathcal{M}'_n = \mathcal{M}_n \cup \{\emptyset\} \]

\[ (***) \qquad \mathcal{M}_1 \neq \{\emptyset\} \quad \text{i.e.} \quad
\exists A \in \mathcal{M}_1,\ A \neq \emptyset \]

\[ \mathcal{M}_\emptyset = \mathfrak{P}(R^\emptyset) = \{\emptyset,
\{e\}\} \]

\[ x = (x_1, \dots, x_n) \in R^n \]

\[ \{x\} \in R^n \iff \forall i \in [1,n], \text{ on a } \{x_i\} \in
\mathcal{M}_1 . \]

\[ R_0 = \{x \in R \mid \{x\} \in \mathcal{M}_1\} \qquad \text{donc } R_0
\subset R , \]

\[ R_0^I = \{x \in R^I \mid \{x\} \in \mathcal{M}_I\} \qquad (R_0^I \subset
R^I) \]

\[ \Gamma_f \subset A \times B \subset R^I \times R^J \simeq R^{I \sqcup J} .
\]

\[ \mathrm{Mod}_{\mathcal{M}_*} \longrightarrow (\mathrm{Ens}) \qquad (I, A)
\mapsto A . \]

\[ \mathrm{Esp}_{\mathcal{M}_*} \longrightarrow (\mathrm{Ens}) \qquad X
\mapsto |X| \]

\[ \begin{cases} X \\ |X| \xrightarrow[\sim]{\ \alpha\ } E \end{cases} \]

\[ (\mathrm{M}5) \qquad R_0 \neq \emptyset \]

\[ (\mathrm{M}0) \qquad R \in \mathcal{M}_1 \qquad (\Rightarrow \forall I \
(\text{ens.\ fini}),\ R^I \in \mathcal{M}_I) \]

\[ R \in \operatorname{Ob} \Sigma \qquad (\Sigma = \mathrm{Esp}_{\mathcal{M}_*})
\]

\[ (\Sigma, R, X \mapsto |X|) \]

\[ X_1 \sqcup_Z X_2 , \]

\[ (\mathrm{M}5) \qquad \forall n,\ \mathcal{M}_n \text{ stable par réunions
finies} \]

\[ \begin{cases} f_\alpha : X_1 \to R \\ g_\alpha : X_2 \to R \end{cases} \]

\[ X_1 \to R^I \qquad (X_2 \to R^A) \]

\[ (f_{\alpha i})_{i \in I} \qquad f_{\alpha i} : X_i \to R \text{
modérée} , \]

\[ f_\alpha : X \to R \]

\[ X_i \times Y \xrightarrow{\ g_i \times \mathrm{id}_Y\ } X \times Y . \]

\[ (\mathrm{M}6) \qquad \begin{cases}
\forall\, Z, X \in \mathcal{M}_n,\ Z \subset X,\ \exists\,
(f_\alpha)_{\alpha \in A} \text{ famille finie de fonctions modérées } X \to
R \\
\text{et des } (e_\alpha)_{\alpha \in A},\ e_\alpha \in R_0, \text{ telles
que } Z = \bigcap f_\alpha^{-1}(e_\alpha) . \\
\text{De plus OPS que les } f_\alpha \text{ séparent les points de } X
\smallsetminus Z
\end{cases} \]

\[ (\mathrm{M}7) \qquad \begin{cases} \forall\, Z, X \in \mathcal{M}_n,\ Z \subset X, \text{
et toute } f : Z \to R \text{ modérée,} \\ \exists\, g : X \to R \text{
modérée qui prolonge } f \end{cases} \]

\[ X_1 \sqcup_Z X_2 \qquad (Z \hookrightarrow X_1,\ Z \hookrightarrow X_2) \]

\[ X_i \longrightarrow R^I \qquad (i \in \lbrace 1, 2 \rbrace) \]

\[ X_i \longrightarrow R^{A_i} \qquad i \in \lbrace 1, 2 \rbrace \]

\[ X_1 \to R^{I \sqcup A_1} \]

\[ A \in \mathcal{M}, \ B \in \mathcal{M}_A \Rightarrow B \in \mathcal{M} .
\]

\[ (\mathrm{M}8) \qquad R \text{ est réunion de ses parties modérées} \]

\[ \forall A \in \mathcal{M}_{\mathfrak{X}}, \text{ on a } A \cap \mathfrak{Y}
\in \mathcal{M}_{\mathfrak{X}} . \]

\[ \mathfrak{X}_1 \sqcup_{\mathfrak{Z}} \mathfrak{X}_2 \]

\[ (\mathrm{M}9) \qquad \begin{cases} R \text{ séparé} \\ \text{les parties
modérées de } R^n \text{ sont compactes} \end{cases} \]

\[ \mathrm{Esp}_{\mathcal{M}_*} \longrightarrow (\text{Esp.\ top.\ cpts}) \]

\[ (\mathrm{M}10) \qquad \text{Dans } R, \text{ tout pt a un syst.\ fond.\
de voisinages} \in \mathcal{M}_1 \]

\[ (\mathrm{M}11) \qquad \begin{cases}
\text{a) } 0 \neq 1 \text{ sont } \in R_0 \text{ i.e.\ } \lbrace 0 \rbrace,
\lbrace 1 \rbrace \in \mathcal{M}_1 \\
\text{b) } \forall A \in \mathcal{M}_2, \text{ les fonctions } (x,y) \mapsto
x - y, \ (x,y) \mapsto xy \\ \qquad \text{sur } A \text{ sont modérées, i.e.\ les} \\
\qquad \lbrace (x, y, x-y) \mid (x,y) \in A \rbrace \text{ et } \lbrace (x,
y, xy) \mid (x,y) \in A \rbrace \\ \qquad \text{sont modérées dans } R^3 \\
\text{c) } \forall A \subset R^*,\ A \in \mathcal{M}_1,\ \exists B \in
\mathcal{M}_1 \text{ avec } A^{-1} \subset B
\end{cases} \]

\[
  \lbrace (x,y) \in R^2 \mid xy = 1,\ x \in A \rbrace .
\]

\[
  Y = \bigcap_{\alpha \in I} f_\alpha^{-1}(\lbrace e_\alpha \rbrace),
  \qquad e_\alpha \in R ,
\]

\[
  \underbrace{f_\alpha}_{\lambda} g_\beta(x) = \lambda\, g_\beta(x)
  \neq f_\alpha g_\beta(y) = \lambda\, g_\beta(y) .
\]

\[
  (Z_i)_{i \in \pi_0(Y \smallsetminus Y')}, \qquad
  \underbrace{\pi_0(Y \smallsetminus Y')}_{I} = (U_i)_{i \in I},
  \qquad Z_i \ \text{modérés}
\]

\[
  R_i \subset (\overline{U}_i \times Z_i) \times X
\]