A survey and exposition of sub-Hausdorff separation axioms

Abstract

This paper surveys and develops sub-Hausdorff axioms. It augments known relationships involving T ₀ (Kolmogorov), S ₀ (quasi-sobriety), S ₁ (sobriety), T ₁ (Fréchet), and T ₂ (Hausdorff ) by examining a large suite of sub-Hausdorff axioms. Emphasis is given to hereditary S ₀, hereditary S ₁, deleted S ₀, deleted S ₁, (locally) strong S ₀, (locally) strong S ₁, T_sc , and pre-T ₂. Properties of the Kolmogorov functor are applied to preservation, reflection, and characterization of certain sub-Hausdorff axioms. Classes of examples are constructed to illustrate main ideas, show non-reversibility of key implications, and correct published errors. Many theorems are given both point-set based and spectrum based proofs for additional insights. Links to domain theory and the topology of digital displays are noted. Key relationships are summarized in two tables.