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We introduce and study the separation axiom h-RT , designed to provide a pointwise measure of the discrepancy between the h-closure and the h-kernel of a singleton in the associated topology τh. Using the specialization preorder of (X, τh), we obtain structural characterizations of h-R0, h-T0, and h-RT . We prove that every h-RT space satisfies a conditional singleton-derivedset property and an h-RH closure-intersection condition, while the conjunction of h-T0 and h-RT implies the global h-RD condition. Explicit finite examples separate h-RT from h-R0, h-T0, and h-RD. We establish preservation results for compatible subspaces and finite products, characterize locally h-indiscrete spaces through the associated topology, and introduce the weakly h-R0 axiom inspired by Di Maio’s weakly R0 condition. For weakly h-R0 spaces, we obtain a kernel characterization, a preservation theorem for injective always h-closed maps, and a product theorem under an explicit compatibility condition on the associated topologies.