The Separation Hierarchy

Topology · Axiom Academy

LESSON The Separation Hierarchy Understanding the complete chain of separation axioms from T₀ to Normal spaces Step 1: The Hierarchy Overview The separation axioms form a beautiful hierarchy of increasingly strong conditions that measure how well a topological space can "separate" points and sets. Each condition in the hierarchy implies all the ones before it, creating a chain of implications. We have the following chain of implications: Each arrow represents a strict implication: every space satisfying a stronger property also satisfies all weaker properties, but the reverse is not always true. Step 2: T₀ (Kolmogorov) Spaces A topological space is T₀ (or Kolmogorov) if for any two distinct points , there exists an open set containing one but not the other. T₀ is the weakest separation axiom. It says we can "topologically distinguish" any two distinct points—at least one of them has a neighborhood that excludes the other. All T₁, T₂, Regular, and Normal spaces (by the hierarchy) The Sierpiński space: with topology Any space with the cofinite topology Most partially ordered sets with the Alexandrov topology The Indiscrete Topology: Let with elements, and give it the indiscrete topology . Then for any two distinct points , every open set contains both points. Thus we cannot separate them, and the space is not T₀. A topological space is T₁ (or Fréchet) if for any two distinct points , there exist open sets and such that and .

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