Definition:Kolmogorov Space/Definition 1

Definition
Let $T = \struct {S, \tau}$ be a topological space.

$\struct {S, \tau}$ is a Kolmogorov space or $T_0$ space :


 * $\forall x, y \in S$ 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 every two elements in $S$ are topologically distinguishable.

Also see

 * Equivalence of Definitions of $T_0$ Space