On h-Separability and Countable h-Dense Homogeneity via the Associated Topology

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.