On h-R^⋆ Topological Spaces via the Associated Topology

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.