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 τh = 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.