Non-Hausdorff Separation Axioms
Tianyi Zhou
TL;DR
This survey provides a cohesive account of non-Hausdorff separation axioms situated between $T_0$, $T_1$, and Hausdorff, emphasizing specialization preorders and their quotients, Skula modifications, sobrification, and intermediate classes like $T_D$, $R_0$, and $R_1$. It develops a unifying framework based on weak/strong (initial/final) topologies, box/product constructions, and lattice-frame theory to analyze and compare these axioms, their interrelations, and their stability under common constructions (quotients, products, and gluing). Key contributions include characterizations of $T_0$, $T_1$, $R_0$, and $R_1$ via preorders and derived sets, a detailed account of Skula topology and its interaction with $T_0$-quotients, and the sober/Quasi-sober/Sobrification story linking point-free and locale-theoretic perspectives. The work also highlights how these properties behave under standard operations and provides a catalog of implications and counterexamples illustrating the landscape between $T_1$ and Hausdorff, with practical relevance to non-Hausdorff manifolds, algebraic geometry, and logic. Overall, the notes offer a structured reference for researchers studying non-Hausdorff separation axioms and their categorical/topological ramifications.
Abstract
This note is an introductory survey of non-Hausdorff separation axioms. The main focus is to study properties that are between $T_0$ and $T_1$, properties between $T_1$ and Hausdorff and how the $T_0$-quotient change them and the relation between them.