International Journal of Topology

We investigate regularity properties defined by means of h-open sets, with systematic use of the associated topology ${\tau }_h=\mathrm{hO}\left(X\right)$. In this framework, h-regularity separates points from $\tau$-closed sets by ${\tau }_h$-open sets, whereas strong h-regularity is precisely ordinary regularity of . After recalling the established characterization of h-regularity, we obtain preservation results under suitable mappings, study the relation with $h^{\ast }$-regularity and extremal h-disconnectedness, analyze strong -continuity, and establish hereditary results for the regularity notions under the appropriate subspace hypotheses. We also connect the h-$R_0$ condition with the associated topology and with strong h-regularity.

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.

The Standard Model of particle physics has achieved unprecedented success in describing fundamental interactions through the formalism of Quantum Field Theory (QFT), where particles are treated as point-like excitations of underlying fields. In this paper, we introduce an alternative, highly elegant mathematical hypothesis: an effective model that describes particle properties not as fundamental points but as 3D projections of complex topological defects originating in a 4D manifold. By applying the geometry of Kaleidoscopic Iterated Function Systems (KIFS) to a zero-viscosity hydrodynamic framework, we demonstrate a formal isomorphism between standard quantum numbers and 4D fractal attractors. This approach seeks not to replace established phenomenological interactions but rather to offer a complementary geometric lens—a “topological translation”—through which mass generation, spin, and localization can be visualized as emergent properties of higher-dimensional fluid mechanics.