Bir $\left( X,\tau \right) $ topolojik uzayının kompakt uzay olması için gerek ve yeter koşul uzayın kapalılar ailesinin sonlu kesişim özelliğine sahip her altailesinin kesişiminin boştan farklı olmasıdır.
Biçimsel olarak
\begin{equation*}
\begin{array}{c}
(X,\tau ),\text{ topolojik uzay} \\
:\Rightarrow \\
(X,\tau ),\text{ kompakt uzay}\Leftrightarrow \forall \mathcal{A}\left[
\left( \mathcal{A}\subseteq \mathcal{C}(X,\tau )\right) \left( \mathcal{A},%
\text{ s.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{ s.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{ s.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{ s.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{ s.k.ö. değil}"
\end{array}
\end{equation*} önermesinin doğru olduğunu göstermek yeterli olacaktır.
$\left( \Rightarrow \right) :$ $\left( X,\tau \right) $ kompakt uzay, $\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}{c}
\left( \mathcal{B}\subseteq \tau \right) \left( X=\setminus \emptyset
=\setminus \left( \cap \mathcal{A}\right) =\cup \mathcal{B}\right) \\
\\
(X,\tau ),\text{ kompakt uzay}\Rightarrow X, \tau\text{-kompakt}
\end{array}
\right\} \Rightarrow
\end{array}
$
$\mbox{}$
$\left.
\begin{array}{r}
\Rightarrow \left(\exists \mathcal{B}^{\ast }\subseteq \mathcal{B}\right)
( \left\vert \mathcal{B}^{\ast }\right\vert <\aleph _{0})(X=\cup
\mathcal{B}^{\ast }) \\
\\
\mathcal{A}^{\ast }:=\left\{ A|\setminus A\in \mathcal{B}^{\ast }\right\}
\end{array}
\right\} \Rightarrow \left( \mathcal{A}^{\ast }\subseteq \mathcal{A}\right)
\left( \left\vert \mathcal{A}^{\ast }\right\vert <\aleph _{0}\right) \left(
\cap \mathcal{A}^{\ast }=\emptyset \right) \!\!\!\!\!$
$\mbox{}$
$\left.
\begin{array}{c}
\Rightarrow \mathcal{A},\text{ s.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{ s.k.ö. değil}\Rightarrow \left(
\exists \mathcal{B}^{\ast }\subseteq \mathcal{B}\right) \left( \left\vert
\mathcal{B}^{\ast }\right\vert <\aleph _{0}\right) \left( \cap \mathcal{B}%
^{\ast }=\emptyset \right) \\
\\
\mathcal{A}^{\ast }:=\left\{ \setminus A|A\in \mathcal{B}^{\ast }\right\}%
\end{array}
\right\} \Rightarrow $
$\mbox{}$
$\left.
\begin{array}{c}
\Rightarrow \left( \mathcal{A}^{\ast }\subseteq \mathcal{A}\right) \left(
\left\vert \mathcal{A}^{\ast }\right\vert <\aleph _{0}\right) \left(
X=\setminus \emptyset =\setminus \left( \cap \mathcal{B}^{\ast }\right) =%
\underset{A\in \mathcal{B}^{\ast }}{\cup }(\setminus A)=\cup \mathcal{A}%
^{\ast }\right) .
\end{array}
\right.$