# Topological indistinguishability

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(Redirected from Topologically distinguishable points)

## Definition

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

- The closures of the singleton sets and are equal.
- Every closed subset containing contains and every closed subset containing contains .
- Every open subset containing contains and every open subset containing contains .

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.