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.