\begin{definition}[Definition:Kolmogorov Space/Definition 1]
Let $T = \struct {S, \tau}$ be a topological space.
$\struct {S, \tau}$ is a '''Kolmogorov space''' or '''$T_0$ space''' {{iff}}:
:$\forall x, y \in S$ such that $x \ne y$, either:
:: $\exists U \in \tau: x \in U, y \notin U$
:or:
:: $\exists U \in \tau: y \in U, x \notin U$
That is:
: for any two distinct elements $x, y \in S$ there exists an open set $U \in \tau$ which contains one of the elements, but not the other.
That is:
:$\struct {S, \tau}$ is a '''$T_0$ space''' {{iff}} every two elements in $S$ are topologically distinguishable.
\end{definition}