\[
\frac{P'(z)}{P(z)} - \frac{Q'(z)}{Q(z)} = 0 ,
\]

\[
Q(z)P'(z) - P(z)Q'(z) = (Q(z)-P(z))\,P'(z) + P(z)\,(P'(z)-Q'(z))
\]

\[
e_i e_1 = 0, \qquad
f_i f_1 = \begin{cases} 0 & \text{si } i \text{ impair}\\
f_{i+1} & \text{si } i \text{ pair}\end{cases}, \qquad
f_i e_1 = e_{i+1}, \qquad
e_i f_1 = \begin{cases} -e_{i+1} & \text{si } i \text{ impair}\\
0 & \text{si } i \text{ pair}\end{cases}
\]

\[
(f_1+\lambda e_1)(f_1+\lambda e_1) = f_1f_1 + \lambda^2 e_1e_1 +
\lambda(f_1e_1 + e_1f_1), \qquad e_2 = f_1e_1, \quad e_3 = f_2e_1 .
\]

\[
e_ie_1 = 0,\qquad f_if_1 = \begin{cases} 0 & i \text{ impair}\\
f_{i+1} & i \text{ pair}\end{cases},\qquad
f_ie_1 = \eta_{i+1} = \alpha e_{i+1} + \beta f_{i+1},
\]

\[
e_if_1 = \begin{cases} -\eta_{i+1} = -\alpha e_{i+1} - \beta f_{i+1} &
i \text{ impair}\\
-\eta_{i+1} + e_{i+1} = (1-\alpha)e_{i+1} - \beta f_{i+1} & i \text{ pair}
\end{cases}
\]

\[
(e_if_1)e_1 = \begin{cases} -\eta_{i+1}e_1 = -\beta f_{i+1}e_1 =
-\beta\eta_{i+2} & i \text{ impair}\\
-\eta_{i+1}e_1 + \ill{} = -\beta\eta_{i+2} & i \text{ pair}\end{cases}
\]

\[
(e_if_1)e_1 = e_i(f_1e_1) = -e_i(e_1f_1) = -(e_ie_1)f_1 = 0
\]

\[
\delta(e_1e_{d-2}) = -e_1\,\delta(e_{d-2}) = -e_1e_{d-3}, \qquad
\delta(a\eta) = a\,\delta\eta = a\alpha e_{d-2} + a\beta f_{d-2},
\]

\[
\delta(e_1f_{d-2}) = -e_1f_{d-3} = b\alpha e_{d-2} + b\beta f_{d-2},\qquad
\delta(f_1e_{d-2}) = \delta(f_1)e_{d-2} - f_1\delta(e_{d-2}) =
e_{d-2} - f_1e_{d-3},
\]

\[
\delta(c\eta) = c\,\delta\eta = c\alpha e_{d-2} + c\beta f_{d-2} ;
\]

\[
\Bigl(\frac{f}{g}\Bigr)' = \frac{f'g-g'f}{g^2}, \qquad
(xy)'' = x''y + 2x'y' + xy'', \qquad
\Bigl(\frac{P}{Q}\Bigr)' = \frac{P'}{Q} - P\frac{Q'}{Q^2} =
\frac{P'Q-PQ'}{Q^2},
\]

\[
\Bigl(\frac{P}{Q}\Bigr)'' = \frac{P''}{Q} - 2P'\frac{Q'}{Q^2} + \ill{}, \qquad
\frac{P''Q^2 - Q''PQ - 2P'QQ' + 2PQ'^2}{Q^3} .
\]

\[
\mathbb{P}(X\times Z) \overset{?}{\simeq}
\mathrm{Hom}(Z\times X,\ \mathcal{E}) =
\mathrm{Hom}(X,\ \underline{\mathrm{Hom}}(Z,\mathcal{E}))
\]

\[
\mathrm{Hom}(\mathcal{E},\ \underline{\mathrm{Hom}}(X,Z)) =
\mathbb{P}(\underline{\mathrm{Hom}}(X,Z))
\]

\[
\underline{\mathrm{Hom}}(X,Y)\subset\mathbb{P}(X\times Y), \qquad
\underline{\mathrm{Hom}}(Z\times X,\ Y)\subset\mathbb{P}(Z\times X\times Y)
\]

\[
X = \Sigma_1^+\sqcup E_1\sqcup\Sigma_1^-, \qquad
X = \Sigma_2^+\sqcup E_2\sqcup\Sigma_2^-
\]

\[
X = \Sigma_1^+\sqcup E_1\sqcup\Sigma_{12}\sqcup E_2\sqcup\Sigma_2^+
\]

\[
\Sigma_{12} \overset{\mathrm{déf}}{=} \Sigma_1^-\cap\Sigma_2^- .
\]

\[
\Sigma_1^+ \subsetneq \overline{\Sigma_1^+} \subset \Sigma_1^+\cup E_1
\]

\[
\overline{\Sigma_{12}} \subset \overline{\Sigma_1^-}\cap
\overline{\Sigma_2^-} \subset (\Sigma_1^-\cup E_1)\cap(\Sigma_2^-\cup E_2)
\subset \Sigma_{12}\cup E_1\cup E_2
\]

\[
\Sigma_2^+ \subsetneq \overline{\Sigma_2^+} \subset \Sigma_2^+\cup E_2
\]

\[
\overline{\Sigma_{12}}\cap E_1\neq\emptyset, \qquad
\overline{\Sigma_{12}}\cap E_2\neq\emptyset
\]

\[
\overline{\Sigma_1^-} = \overline{\Sigma_2^+\sqcup E_2\sqcup\Sigma_{12}} =
\overline{\Sigma_2^+}\cup E_2\cup\overline{\Sigma_{12}} \subset
\Sigma_1^-\cup\overline{\Sigma_{12}}
\]

\[
\overline{\Sigma_i^+} = \Sigma_i^+\cup E_i, \qquad
\overline{\Sigma_i^-} = \Sigma_i^-\cup E_i .
\]

\[
\overline{\Sigma_{12}} = \Sigma_{12}\cup E_1\cup E_2
\]

\[
X = E_j \sqcup \coprod_{\alpha\in\pi_0(X\setminus E_j)} \Sigma_j^{\alpha}
\]

\[
U_{J,i}\cap\Sigma_i^+, \qquad U_{J,i}\cap\Sigma_i^-
\]

\[
U_{J,i} = X\setminus\coprod_{\alpha\in K_0}\widehat{\Sigma}_\alpha^+
\]

\[
\begin{cases}
K_0\subset J\\
\Sigma_\alpha^+\in\pi_0(X\setminus E_\alpha) \quad (\forall\alpha\in K_0)\\
\widehat{\Sigma}_\alpha^+ = \Sigma_\alpha^+\cup E_\alpha .
\end{cases}
\]

\[
U = U_{J,i}\cap\Sigma_i^- = X\setminus\Bigl(\coprod_{\alpha\in K_0}
\widehat{\Sigma}_\alpha^+ \cup \widehat{\Sigma}_i^+\Bigr)
\]

\[
U = X\setminus\Bigl(\bigcup_{\alpha\in K_1}\widehat{\Sigma}_\alpha^+
\cup\widehat{\Sigma}_i^+\Bigr)
\]

\[
U_{J,i}\cap\Sigma_i^+, \qquad U_{J,i}\cap\Sigma_i^-
\]

\[
X\setminus\bigcup_{\alpha\in K}(\Sigma_\alpha^+\cup E_\alpha)
\]

\[
\widehat{\Sigma}_{\alpha_0}\cap\widehat{\Sigma}_i^+ = \emptyset \quad
\text{i.e.}\quad \Sigma_{\alpha_0}\cap\Sigma_i = \emptyset
\]

\[
U = X\setminus\bigcup_{\alpha\in K}\widehat{\Sigma}_\alpha^+ \qquad
\bigl|\ K = K_1\cup\{i\}\ \struck{\ill{}}
\]

\[
E_i\subset U_{J,i} = X\setminus\bigcup_{\alpha\in K_0}
\widehat{\Sigma}_\alpha^+
\]

\[
E_i\subset X\setminus\widehat{\Sigma}_{\alpha_0}^+ = \Sigma_{\alpha_0}^-
\]

\[
\Sigma_{\alpha_0}^+\not\subset\Sigma_i^+
\]

\[
V = X\setminus\bigcup_{\alpha\in K}\widehat{\Sigma}_\alpha^+
\]

\[
\begin{cases}
K = \{\alpha\in I \mid \overline{V}\cap E_\alpha\neq\emptyset \ \text{ i.e. }\ E_\alpha\cap\dot{V}\neq\emptyset\}\\
\Sigma_\alpha^+ \text{ est l'unique comp.\ de } X\setminus E_\alpha
\text{ contenue dans } X\setminus V
\end{cases}
\]

\[
  \left\{
  \begin{array}{l}
    \overline{V}\cap E_{\alpha}\neq\emptyset\\
    V\cap\Sigma_{\alpha}^{-}\neq\emptyset\quad\text{dès que}\quad
    V\cap\Sigma_{\alpha}^{+}=\emptyset
  \end{array}
  \right.
\]

\[
  \overline{V}\cap E_{\alpha}=\overline{\Sigma_{\alpha}^{-}}\cap E_{\alpha}
  \qquad(\text{qui est donc}\neq\emptyset)
\]

\[
  Y=\widehat{\Sigma}_{\alpha}^{+},\qquad
  Z=\bigcup_{\beta\in K\smallsetminus\{\alpha\}}\widehat{\Sigma}_{\beta}^{+}
\]

\[
  \overline{\bigl(X\smallsetminus(Y\cup Z)\bigr)}\cap Y
  =\overline{(X\smallsetminus Y)}\cap Y
\]

\[
  \overline{V}\cap E_{\beta}=\emptyset .
\]

\[
  E_{\beta}\subset X\smallsetminus V=\bigcup_{\alpha\in
  K}\widehat{\Sigma}_{\alpha}^{+},
\]

\[
  E_{\beta}\subset\bigcup\Sigma_{\alpha}^{+}\qquad(\text{ouvert disjoint de
  }V)
\]

\[
  \alpha\neq\beta\in K,\ \alpha\neq\beta\ \Rightarrow\ \Sigma_{\alpha}^{+}\cap\Sigma_{\beta}^{+}=\emptyset .
\]

\[
  V=X\smallsetminus\bigcup_{\alpha\in K}\widehat{\Sigma}_{\alpha}^{+} .
\]

\[
  (s,i)\in R\ \overset{\text{déf}}{\Longleftrightarrow}\ \overline{U_{s}}\cap E_{i}\neq\emptyset .
\]

\[
  U_{s}=X\smallsetminus\bigcup_{\alpha\in K}\widehat{\Sigma}_{\alpha}^{+}
  \qquad\text{où } i\in K\subset I
\]

\[
  K\smallsetminus\{i\}=\{\alpha\in I\mid E_{\alpha}\subset\Sigma_{\alpha}^{-}\}
\]

\[
  (\Sigma^{\alpha})_{\alpha\in\omega}\qquad(\text{les “hémisphères”})
\]

\[
  \Sigma_{1}^{+}\cap\Sigma_{2}^{+}=\emptyset\qquad(\text{d'où }E_{1}\cap
  E_{2}=\emptyset)
\]

\[
  \begin{array}{ll}
    \Sigma_{1}^{+}\subset\Sigma_{2}^{-} &
    \text{i.e.}\ \Sigma_{1}^{+}\cap\Sigma_{2}^{-}=\Sigma_{1}^{+}\\
    \Sigma_{2}^{+}\subset\Sigma_{1}^{-} &
    \text{i.e.}\ \Sigma_{2}^{+}\cap\Sigma_{1}^{-}=\Sigma_{2}^{+}
  \end{array}
\]

\[
  \Sigma_{12}\overset{\text{déf}}{=}\Sigma_{1}^{-}\cap\Sigma_{2}^{-}\supset
  E_{1}\cup E_{2},\qquad\Sigma_{1}^{-}\cap\Sigma_{2}^{-}\neq\emptyset
\]

\[
  \Sigma_{1}^{+}\supset E_{1}\hookrightarrow\Sigma_{12}\hookleftarrow
  E_{2}\hookrightarrow\Sigma_{2}^{+}
\]

\[
  X=\coprod_{s\in S}U_{s}\ \amalg\ \coprod_{a\in A}E_{a} .
\]

\[
  X'=\coprod_{s\in S'}U_{s}\amalg\coprod_{a\in A'}E_{a},\qquad
  X''=\coprod_{s\in S''}U_{s}\amalg\coprod_{a\in A''}E_{a},
\]

\[
  X'\cap X''=E_{a_{0}} .
\]

\[
  \mathcal{X}=(\boldsymbol{\Gamma},S)
\]

\[
  \mathcal{X}^{*}=(\boldsymbol{\Gamma}^{*},\underbrace{S\amalg A'}_{S^{*}}),
\]

\[
  U_{s}\neq\emptyset\quad\text{ou}\quad E_{a}\neq\emptyset .
\]

\[
  \Pi=(\Sigma^{\varepsilon})_{\varepsilon\in\omega}\qquad(\omega\text{ ens.\ à deux éléments})
\]

\[
  X=\bigcup_{\varepsilon\in\omega}\Sigma^{\varepsilon} .
  \tag{1}
\]

\[
  \Sigma^{\varepsilon}\neq\emptyset\qquad(\forall\varepsilon\in\omega)
  \tag{2}
\]

\[
  \Pi\neq\{\emptyset,X\}
  \tag{2'}
\]

\[
  \Sigma^{\varepsilon}\neq\Sigma^{\varepsilon'}\quad\text{si}\quad
  \varepsilon,\varepsilon'\in\omega,\ \varepsilon\neq\varepsilon' .
  \tag{3}
\]

\[
  \Sigma^{\varepsilon'}\subset\Sigma^{\varepsilon}
\]

\[
  \Sigma^{\varepsilon}=X .
\]

\[
  E=\bigcap_{\varepsilon\in\omega}\Sigma^{\varepsilon}
  \tag{4}
\]

\[
  \left\{
  \begin{array}{l}
    X\ \text{connexe}\ \Longrightarrow\ E\overset{\text{déf}}{=}
    \bigcap_{\varepsilon\in\omega}\Sigma^{\varepsilon}\neq\emptyset\\
    \text{Si les }\Sigma^{\varepsilon}\text{ connexes, alors }
    X\ \text{connexe}\ \Longleftrightarrow\ E\neq\emptyset .
  \end{array}
  \right.
  \tag{5}
\]

\[
  \Sigma^{0\varepsilon}=\Sigma^{\varepsilon}\smallsetminus E
  =\Sigma^{\varepsilon}\smallsetminus\Sigma^{\varepsilon}\cap
  \Sigma^{\varepsilon'}=X\smallsetminus\Sigma^{\varepsilon'}
  \qquad(\omega=\{\varepsilon,\varepsilon'\})
  \tag{6}
\]

\[
  X=\Sigma^{0\varepsilon_{1}}\cup E\cup\Sigma^{0\varepsilon_{2}}
  \qquad(\omega=\{\varepsilon_{1},\varepsilon_{2}\})
  \tag{7}
\]

\[
  \overline{\Sigma^{0\varepsilon}}\cap E=\overline{\Sigma^{0\varepsilon}}\cap
  \Sigma^{\varepsilon'}=\overline{\Sigma^{0\varepsilon}}\smallsetminus
  \Sigma^{0\varepsilon}=\dot{\Sigma}^{0,\varepsilon} ,
  \tag{8}
\]

\[
  \Pi_{1}=\{\Sigma_{1}^{\varepsilon}\}_{\varepsilon\in\omega_{1}},\qquad
  \Pi_{2}=\{\Sigma_{2}^{\varepsilon}\}_{\varepsilon\in\omega_{2}}
  \tag{9}
\]

\[
  \Sigma_{1}^{+}\cap\Sigma_{2}^{+}=\emptyset
  \tag{10}
\]

\[
  \Sigma_{1}^{+}\subset\Sigma_{2}^{-}
  \tag{11}
\]

\[
  \Sigma_{2}^{+}\subset\Sigma_{1}^{-}
  \tag{11 bis}
\]

\[
  \Pi_{1}=\{\Sigma_{1}^{+},\Sigma_{1}^{-}\},\qquad
  \Pi_{2}=\{\Sigma_{2}^{+},\Sigma_{2}^{-}\}
  \tag{12}
\]

\[
  \Sigma_{12}=\Sigma_{1}^{-}\cap\Sigma_{2}^{-},\qquad
  E_{1}=\Sigma_{1}^{+}\cap\Sigma_{1}^{-},\qquad
  E_{2}=\Sigma_{2}^{+}\cap\Sigma_{2}^{-}
  \tag{13}
\]

\[
  \left|
  \begin{array}{l}
    E_{1}=\Sigma_{1}^{+}\cap\Sigma_{12},\quad
    \Sigma_{2}^{-}=\Sigma_{1}^{+}\cup\Sigma_{12}
  \end{array}
  \right.
\]

\[
  \left\{
  \begin{array}{l}
    E_{2}=\Sigma_{12}\cap\Sigma_{2}^{+},\quad
    \Sigma_{1}^{-}=\Sigma_{12}\cup\Sigma_{2}^{+}\\[4pt]
    \Sigma_{12}\overset{(13)}{=}\Sigma_{1}^{-}\cap\Sigma_{2}^{-},\quad
    X=\Sigma_{2}^{-}\cup\Sigma_{1}^{-} .
  \end{array}
  \right.
  \tag{15}
\]

\[
  \left\{
  \begin{array}{ll}
    \text{a)} & \Sigma_{1}^{+}\cap\Sigma_{2}^{-}=\Sigma_{1}^{+}\neq\emptyset\\
    \text{b)} & \Sigma_{2}^{+}\cap\Sigma_{1}^{-}=\Sigma_{2}^{+}\neq\emptyset\\
    \text{c)} & \Sigma_{1}^{-}\cap\Sigma_{2}^{-}=\Sigma_{12}\neq\emptyset ,
  \end{array}
  \right.
  \tag{16}
\]

\[
  \Sigma_{1}^{-}=X\smallsetminus\Sigma_{1}^{+},\qquad
  \Sigma_{2}^{-}=X\smallsetminus\Sigma_{2}^{+}
\]

\[
  \Sigma_{12}=X\smallsetminus(\Sigma_{1}^{+}\cup\Sigma_{2}^{+})
\]

\[
  X=\Sigma_{1}^{+}\amalg\Sigma_{2}^{+}
\]

\[
  \Pi_{1}=\{\Sigma_{1}^{+},\underset{\Sigma_{1}^{-}=X\smallsetminus
  \Sigma_{1}^{+}}{\underset{\shortparallel}{\Sigma_{2}^{+}}}\},\qquad
  \Pi_{2}=\{\Sigma_{2}^{+},\Sigma_{1}^{+}\}
\]

\[
  (\Sigma_{1}^{+},\Sigma_{2}^{+})\in\Pi_{1}\times\Pi_{2}
\]

\[
  \Sigma_{1}^{+}\cap\Sigma_{2}^{+}=\emptyset .
\]

\[
  C=\Pi_{1}\amalg\Pi_{2}=\omega_{1}\amalg\omega_{2}
  \tag{17}
\]

\[
  C\longrightarrow\mathfrak{P}_{\text{fermé}}(X)
  \tag{18}
\]

\[
  \{\Sigma_{1}^{+},\Sigma_{1}^{-},\Sigma_{2}^{+},\Sigma_{2}^{-}\},
  \tag{19}
\]

\[
  \Sigma_{1}^{+},\ \Sigma_{1}^{-},\ \Sigma_{2}^{+},\ \Sigma_{2}^{-}
\]

\[
  \Sigma_{1}^{+}\neq\Sigma_{1}^{-},\qquad\Sigma_{2}^{+}\neq\Sigma_{2}^{-}
  \tag{$*$}
\]

\[
  \Sigma_{1}^{+}\neq\Sigma_{2}^{+}
  \tag{$**$}
\]

\[
  \left.
  \begin{array}{ll}
    \text{a)} & (\Sigma_{1}^{+}=\Sigma_{2}^{-})\Longleftrightarrow
    E_{1}=\Sigma_{12}\qquad(\Rightarrow E_{2}=\emptyset)\\
    \text{b)} & (\Sigma_{2}^{+}=\Sigma_{1}^{-})\Longleftrightarrow
    E_{2}=\Sigma_{12}\qquad(\Rightarrow E_{1}=\emptyset)
  \end{array}
  \right.
  \tag{$*{*}*$}
\]

\[
  (\Sigma_{1}^{-}=\Sigma_{2}^{-})\Longleftrightarrow
  (\Sigma_{1}^{-}=\Sigma_{2}^{-}=X)\quad\text{i.e.}\quad\Sigma_{12}=X
  \tag{$*{*}{*}*$}
\]

\[
  \left\{
  \begin{array}{lll}
    \text{a)} & E_{1}=\Sigma_{12} & (\Rightarrow E_{2}=\emptyset)\\
    \text{a')} & E_{2}=\Sigma_{12} & (\Rightarrow E_{1}=\emptyset)\\
    \text{b)} & \Sigma_{12}=X\ \text{i.e.}\ \Sigma_{1}^{-}=\Sigma_{2}^{-}=X &
    (\text{d'où }E_{1}=\Sigma_{1}^{+},\ E_{2}=\Sigma_{2}^{+})
  \end{array}
  \right.
  \tag{20}
\]

\[
  (*)\quad\left\{
  \begin{array}{l}
    \Sigma_{1}^{+}\subset\Sigma_{2}^{-}\\
    \Sigma_{2}^{+}\subset\Sigma_{1}^{-} .
  \end{array}
  \right.
\]

\[
  \Sigma_{1}^{+}\subset\Sigma_{1}^{-}\quad\text{ou}\quad
  \Sigma_{1}^{-}\subset\Sigma_{1}^{+},
\]

\[
  \Sigma_{1}^{-}\neq X,\ \Sigma_{2}^{-}\neq X\ \ (\text{on dit donc que
  }\Pi_{1},\Pi_{2}\text{ sont \emph{propres}}),\quad
  E_{1}\neq\Sigma_{12},\ E_{2}\neq\Sigma_{12} .
  \tag{21}
\]

\[
  C=\Pi_{1}\amalg\Pi_{2}=\omega_{1}\amalg\omega_{2}=\{\varepsilon_{1}^{+},
  \varepsilon_{1}^{-},\varepsilon_{2}^{+},\varepsilon_{2}^{-}\}
  \tag{17}
\]

\[
  \varepsilon_{1}^{+}<\varepsilon_{2}^{-},\qquad
  \varepsilon_{2}^{+}<\varepsilon_{1}^{-}
  \tag{22}
\]

\[
  (\Pi_{\alpha})_{\alpha\in A},\qquad
  \Pi_{\alpha}=\{\Sigma_{\alpha}^{\varepsilon}\}_{\varepsilon\in\omega_{\alpha}}
  \tag{23}
\]

\[
  \forall\alpha,\beta\in A,\ \alpha\neq\beta,\quad\Pi_{\alpha},\Pi_{\beta}
  \text{ sont mut\supplied{uellemen}t non parallèles.}
  \tag{24}
\]

\[
  C=\coprod_{\alpha\in A}\omega_{\alpha}
  \tag{25}
\]

\[
  \Bigl(x<y,\ x\in\omega_{\alpha},\ y\in\omega_{\beta}\Bigr)
  \Longleftrightarrow
  \Bigl(\alpha\neq\beta,\ x\underset{\{\Pi_{\alpha},\Pi_{\beta}\}}{<}y\Bigr)
  \tag{26}
\]

\[
  \begin{array}{c}
    C\overset{\varphi}{\longrightarrow}\mathfrak{P}_{\text{fermé}}(X)\\
    (\alpha,x)\longmapsto\Sigma_{\alpha}^{x}
  \end{array}
  \tag{27}
\]

\[
  C_{\geqslant x}=\{y\in C\mid y\geqslant x\},\qquad
  C_{\leqslant x}=\{y\in C\mid y\leqslant x\}
\]

\[
  \text{si } x<y,\ x<z,\ \text{avec } y\neq z,\ \text{alors } (y,z)\ \text{comparables.}
\]

\[
  (s_{0},i)\longmapsto\{s_{0},us_{0},\dots,u^{i-1}s_{0}\}.
\]

\[
  A^{*}=A\smallsetminus\bigcup_{\substack{B\in\Sigma\\ B\subset A,\ B\neq A}}B .
\]

\[
  \begin{cases}
    R=S\smallsetminus\bigcup_{A\in\Sigma}A^{*}, &
    \widehat{\Sigma}=\Sigma\cup\{R\}\\[2pt]
    R^{*}=R\smallsetminus\bigcup_{A\in\widehat{\Sigma},\ A\subsetneq R}A=R &
  \end{cases}
\]

\[
  \widehat{\Sigma}=
  \begin{cases}
    \Sigma & \text{si } R=\emptyset\\
    \Sigma\cup\{R\} & \text{si } R\neq\emptyset
  \end{cases}
\]

\[
  \begin{array}{ccccc}
    \Sigma_{1} & \Sigma_{2} & \Sigma_{3} & \cdots & \Sigma_{i}\\
    T_{1} & T_{2} & T_{3} & & T_{i}
  \end{array}
\]

\[
  T_{1}=\bigcup_{A\in\Sigma_{1}}A,\qquad S_{1}=S\smallsetminus T_{1}
\]

\[
  T_{2}=T_{1}\cup\bigcup_{A\in\Sigma_{2}'}A,\qquad
  S_{2}=S\smallsetminus T_{2},\qquad
  \operatorname{card}\pi_{0}(S_{2})\neq1
\]

\[
  A^{*}=A\smallsetminus\bigcup_{\substack{B\in\Sigma\\ B\subsetneq A}}B
  \qquad(\supset\partial A\text{, donc }A^{*}\neq\emptyset)
\]

\[
  I'=\bigcup_{A\in\Sigma}A=\bigcup_{A\in\Sigma}A^{*},\qquad
  R=I\smallsetminus I',\qquad R^{*}=R
\]

\[
  \widehat{\Sigma}=
  \begin{cases}
    \Sigma & \text{si } R=\emptyset\\
    \Sigma\cup\{R\} & \text{si } R\neq\emptyset
  \end{cases}
\]

\[
  \widehat{\Sigma}^{*}=\{A^{*}\mid A\in\widehat{\Sigma}\}
\]

\[
  I'=\bigcup_{A\in\Sigma_{\max}}A
\]

\[
  f=f'+f''-2,\qquad g=g'+g''
\]

\[
  f^{*}=f'+f''-2,\qquad g=g'+g''
\]

\[
  f=f'+f'',\qquad g=g'+g''-1
\]