More on h-Open Sets

We study the topology associated with the family of h-open sets and use it to organize several separation properties in a unified way. After recalling the classical closure, kernel and separation axioms, we distinguish the original topology τ from the associated topology τ_h = hO(X) and introduce the h-specialization preorder and the h-kernel. The axiom h-R₀ is characterized as the natural symmetry property of this preorder and by the equality of singleton h-closures and h-kernels. We prove the relations h-T₁ ⇔ h-T₀ + h-R₀ and h-T₂ ⇔ h-T₀ + h-R₁, and characterize h-R₁ by singleton h-θ-closures. We then develop the theory of h-difference sets and the axioms h-D₀, h-D₁ and h-D₂, proving h-D₀ ⇔ h-T₀ and h-D₁ ⇔ h-D₂. A corrected characterization in terms of h-neat points and several preservation results under h-irresolute mappings are obtained. Finally, singleton h-derived sets and h-semisimplicity are related to strong h-regularity.