Separation Axioms Summary

Topology · Axiom Academy

Unit 6 Review: The Hierarchy of Separation Properties Separation axioms form the backbone of point-set topology, providing a hierarchy of increasingly strong conditions that describe how well a topological space can distinguish between points and closed sets. This summary consolidates the key definitions, relationships, major theorems, and essential examples from Unit 6. Intuition: Topology can distinguish points Equivalent: All singleton sets are closed Intuition: Points can be separated by disjoint neighborhoods Critical: Limits of sequences are unique Intuition: Points separated by closures Stronger than Hausdorff, weaker than Regular Hausdorff Intuition: Points and closed sets can be separated Tychonoff (Completely Regular) Intuition: Points and closed sets separated by continuous functions Intuition: Disjoint closed sets can be separated Enables: Urysohn's Lemma and Tietze Extension A topological space is normal if and only if, for any two disjoint closed sets and , there exists a continuous function such that for all and for all . Let be a normal space, a closed subspace, and a continuous function to (or ). Then there exists a continuous extension such that for all . Key Examples and Counterexamples Implications Between Properties The separation axioms form a strict hierarchy: . Each property implies the previous ones, but not conversely. Hausdorff ( ) spaces are the most commonly studied because they ensure unique limits of convergent sequences and nets.

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