Topological indistinguishability

Definition
Topological indistinguishability is an equivalence relation on any topological space. For a topological space $$X$$, two (possibly equal, possibly distinct) points $$x,y \in X$$ are termed topologically indinstinguishable if the following equivalent conditions hold:


 * 1) The closures of the singleton sets $$\{ x \}$$ and $$\{ y \}$$ are equal.
 * 2) Every closed subset containing $$x$$ contains $$y$$ and every closed subset containing $$y$$ contains $$x$$.
 * 3) Every open subset containing $$x$$ contains $$y$$ and every open subset containing $$y$$ contains $$x$$.

Two distinct points that are not topologically indistinguishable are termed topologically distinguishable.

Related notions

 * The Kolmogorov quotient of a topological space is the quotient by the equivalence relation of topological indistinguishability, and it is given the T0 topology.