Söz konusu önerme Lindelöf uzayların bir karakterizasyonudur.
Bir \left( X,\tau \right) topolojik uzayının Lindelöf uzayı olması için gerek ve yeter koşul uzayın kapalılar ailesinin sayılabilir kesişim özelliğine (say.k.ö.) sahip her altailesinin kesişiminin boştan farklı olmasıdır.
Biçimsel olarak
\begin{equation*}
\begin{array}{c}
(X,\tau ),\text{ topolojik uzay} \\
:\Rightarrow \\
(X,\tau ),\text{ Lindelöf uzayı}\Leftrightarrow \forall \mathcal{A}\left[
\left( \mathcal{A}\subseteq \mathcal{C}(X,\tau )\right) \left( \mathcal{A},%
\text{ say.k.ö.}\right) \Rightarrow \cap \mathcal{A}\neq \emptyset \right]%
\end{array}%
\end{equation*}
şeklinde ifade edilir.
\begin{equation*}
\begin{array}{c}
\underset{p}{\underbrace{\left( \mathcal{A}\subseteq \mathcal{C}(X,\tau
)\right) }}\underset{q}{\underbrace{\left( \mathcal{A},\text{ say.k.ö.}%
\right) }}\Rightarrow \underset{r}{\underbrace{\cap \mathcal{A}\neq
\emptyset }}
\end{array}%
\end{equation*} ve
\begin{equation*}
\begin{array}{c}
\left( p\wedge q\right) \Rightarrow r\equiv \left( p\wedge r^{\prime
}\right) \Rightarrow q^{\prime }
\end{array}
\end{equation*} olduğundan
\begin{equation*}
\begin{array}{c}
``\left( \mathcal{A}\subseteq \mathcal{C}(X,\tau )\right) \left( \mathcal{A},
\text{ say.k.ö.}\right) \Rightarrow \cap \mathcal{A}\neq \emptyset "
\end{array}
\end{equation*}önermesi ile
\begin{equation*}
\begin{array}{c}
``\left( \mathcal{A}\subseteq \mathcal{C}(X,\tau )\right) \left( \cap
\mathcal{A}=\emptyset \right) \Rightarrow \mathcal{A},\text{ say.k.ö. değil}"
\end{array}
\end{equation*} önermesi denk önermelerdir. Dolayısıyla
\begin{equation*}
\begin{array}{c}
\left( \mathcal{A}\subseteq \mathcal{C}(X,\tau)\right) \left( \cap \mathcal{A}=\emptyset \right) \Rightarrow \mathcal{A},
\text{ say.k.ö. değil}"
\end{array}
\end{equation*} önermesinin doğru olduğunu göstermek yeterli olacaktır.
\left( \Rightarrow \right) : \left( X,\tau \right) Lindelöf uzayı, \mathcal{A}\subseteq \mathcal{C}\left( X,\tau \right) ve \cap
\mathcal{A}=\emptyset olsun.
\left.
\begin{array}{r}
\left( \mathcal{A}\subseteq \mathcal{C}(X,\tau )\right) \left( \cap \mathcal{A}=\emptyset \right) \\
\\
\mathcal{B}:=\left\{ \setminus A|A\in \mathcal{A}\right\}%
\end{array}
\right\} \Rightarrow \!\!\!
\begin{array}{c}
\mbox{} \\
\mbox{} \\
\left.
\begin{array}{r}
\left( \mathcal{B}\subseteq \tau \right) \left( X=\setminus \emptyset
=\setminus \left( \cap \mathcal{A}\right) =\cup \mathcal{B}\right) \\
\\
(X,\tau ),\text{ Lindelöf uzayı}
\end{array}
\right\} \Rightarrow
\end{array}
\mbox{}
\left.
\begin{array}{r}
\Rightarrow \left(\exists \mathcal{B}^{\ast }\subseteq \mathcal{B}\right)
( \left\vert \mathcal{B}^{\ast }\right\vert \leq\aleph _{0})(X=\cup
\mathcal{B}^{\ast }) \\
\\
\mathcal{A}^{\ast }:=\left\{ A|\setminus A\in \mathcal{B}^{\ast }\right\}
\end{array}
\right\} \Rightarrow \left( \mathcal{A}^{\ast }\subseteq \mathcal{A}\right)
\left( \left\vert \mathcal{A}^{\ast }\right\vert \leq\aleph _{0}\right) \left(
\cap \mathcal{A}^{\ast }=\emptyset \right) \!\!\!\!\!
\mbox{}
\left.
\begin{array}{c}
\Rightarrow \mathcal{A},\text{ say.k.ö. değil.}
\end{array}
\right.
\mbox{}
\left( \Leftarrow \right) : \mathcal{A}\subseteq \tau ve X=\cup \mathcal{A} yani \mathcal{A} ailesi, X kümesinin bir \tau-açık örtüsü olsun.
\left.
\begin{array}{r}
\left( \mathcal{A}\subseteq \tau \right) \left( X=\cup \mathcal{A}\right) \\ \\
\mathcal{B}:=\left\{ \setminus A|A\in \mathcal{A}\right\}
\end{array}\right\} \Rightarrow \begin{array}{c}
\mbox{} \\
\mbox{} \\
\left.
\begin{array}{r}
\left( \mathcal{B}\subseteq \mathcal{C}(X,\tau )\right) \left( \cap \mathcal{B}=\setminus \left( \cup \mathcal{A}\right) =\setminus X=\emptyset \right)
\\
\\
\text{Hipotez}
\end{array}
\right\} \Rightarrow\end{array}
\mbox{}
\left.
\begin{array}{r}
\Rightarrow \mathcal{B},\text{ say.k.ö. değil}\Rightarrow \left(
\exists \mathcal{B}^{\ast }\subseteq \mathcal{B}\right) \left( \left\vert
\mathcal{B}^{\ast }\right\vert \leq\aleph _{0}\right) \left( \cap \mathcal{B}%
^{\ast }=\emptyset \right) \\
\\
\mathcal{A}^{\ast }:=\left\{ A|\setminus A\in \mathcal{B}^{\ast }\right\}%
\end{array}
\right\} \Rightarrow
\mbox{}
\left.
\begin{array}{c}
\Rightarrow \left( \mathcal{A}^{\ast }\subseteq \mathcal{A}\right) \left(
\left\vert \mathcal{A}^{\ast }\right\vert \leq\aleph _{0}\right) \left(
X=\setminus \emptyset =\setminus \left( \cap \mathcal{B}^{\ast }\right) =%
\underset{A\in \mathcal{B}^{\ast }}{\cup }(\setminus A)=\cup \mathcal{A}%
^{\ast }\right) .
\end{array}
\right.