We introduce and investigate forms of regularity and normality defined by means of h-open sets. Using the associated topology τ_h = hO(X), we interpret h-regularity as a relative regularity property in which points and closed sets of the original topology are separated by τ_h-open sets, whereas strong h-regularity is precisely the ordinary regularity of (X, τ_h). We establish characterizations of h-regularity, study its preservation under suitable mappings, compare the new notions with their classical counterparts, and examine related separation properties.

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.

Let (X,τ) be a topological space and let τ_h(X) denote the topology formed by its h-open subsets. We study the separation axiom h-R^⋆ through an intrinsic formulation in the associated space (X,τ_h(X)), using cover-boundedness in its trace subspaces. Relative cover-boundedness is shown to imply global h-boundedness. We prove the strict hierarchy h-R₁ ⇒ h-R^⋆ ⇒ h-R₀ and give explicit examples showing that neither implication reverses. The examples arise from two natural h-fixed topologies obtained by isolating one point in cofinite- and cocountable-type spaces. We also show that strong h-regularity implies h-R^⋆, characterize h-submaximality through the associated topology, and provide an explicit h-submaximal example. A classical heredity result yields heredity for h-subspace compatible subspaces. Under h-product compatibility, arbitrary products of h-R₁ spaces and arbitrary products of strongly h-regular spaces are h-R^⋆. Finally, we establish invariance under h-homeomorphisms and introduce a ≪_h-lifting condition that gives a surjective preservation theorem. Perfect maps between the associated spaces automatically have this lifting property, and a compact inverse image hypothesis provides a further sufficient criterion.

Let (X,τ) be a topological space and let τ_h(X) denote the topology formed by the h-open subsets of X. We develop h-density, h-separability and countable h-dense homogeneity systematically through the associated space (X, τ_h(X)). We prove that h-density and h-separability are precisely ordinary density and separability in the associated topology, and that h homeomorphisms are exactly homeomorphisms between associated spaces. Consequently, (X,τ) is h-countable dense homogeneous (h-CDH) if and only if (X, τ_h(X)) is countable dense homogeneous (CDH). A classical theorem on CDH spaces then yields that every h-CDH space is h-T₁. Hence its h-specialization preorder is equality, hCl({x}) = hKer({x}) = {x} for every point, and the pointwise closure–kernel defect measured by the h-RT condition vanishes identically. We give explicit examples showing that or dinary CDH and h-CDH are incomparable in general, while they coincide in Hausdorff spaces. We formulate a product-transfer principle for arbitrary families under compatibility of associated and product topologies and show, using a metrizable counterexample, that compatibility alone does not make h-CDH productive. Finally, we relate the theory to h-normality, including an infinite-product transfer criterion, and formulate several questions concerning normality, product compatibility and converses to the h-RT consequence.

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 $\tau$ from the associated topology ${\tau }_h=\mathrm{hO}\left(X\right)$ and introduce the h-specialization preorder and the h-kernel. The axiom h-$R_0$ 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-$T_1\iff h$-$T_0+h$-$R_0$ and h-$T_2\iff h$-$T_0+h$-$R_1$, and characterize h-$R_1$ by singleton h-$\theta$-closures. We then develop the theory of h-difference sets and the axioms h-$D_0$, h-$D_1$ and h-$D_2$, proving h-$D_0\iff h$-$T_0$ and h-$D_1\iff h$-$D_2$. 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.

Recently, Abbas introduced the notion of an h-open set as a class of generalized open sets in a topological space. The subsequent corrigendum and addendum of Sharma, Saproo, Billawria and Digra clarified the theory and established that the family of all h-open sets is itself a topology, denoted by τ_h, without any T_1/2 assumption. Motivated by this topological interpretation, we introduce and study the separation axioms h-D₁ and h-D₂ by means of h-difference sets. We prove that these two axioms coincide, relate them to the classical D_i axioms in the associated topology (X, τ_h), examine their interaction with h-T_i and h-symmetric spaces, and obtain preservation results under h-irresolute and h-continuous mappings. Finite and infinite examples are included to distinguish the original topology from the associated topology of h-open sets. We also correct the overly strong assertion that h-symmetry alone implies h-T₁: the additional h-T₀ hypothesis is essential. In addition, we place the h-R₀, h-R_H and h-R_D conditions in the associated-topology framework, establish a compatible subspace theorem, and clarify the interaction between h-compactness and h-T₂. Three vector diagrams summarize the structural relationships among the separation axioms considered.