Kolmogorov space

Definition
A topological space $$X$$ is termed a $$T_0$$ space or Kolmogorov space if it satisfies the following equivalent conditions:


 * 1) For any two distinct points $$x,y \in X$$, there is either an open subset containing $$x$$ but not $$y$$, or an open subset containing $$y$$ but not $$x$$.
 * 2) For any two distinct points $$x,y \in X$$, there is either a closed subset containing $$x$$ but not $$y$$, or a closed subset containing $$y$$ but not $$x$$.
 * 3) For any two distinct points $$x,y \in X$$ such that $$\overline{\{ x \} } \ne \overline{ \{ y \} }$$, i.e., the closures of any two distinct one-point subsets must be distinct.