Topological indistinguishability

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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.

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