Separation Axioms Beyond T2: Novel Properties and Interactions in Non-Regular Spaces

This paper investigates separation axioms that are weaker than the classical Hausdorff condition, examining their relationships in non-regular topological spaces. We focus on spaces that exhibit structural properties characteristic of higher separation axioms through mod-est separation requirements. The study establishes new characterizations of sigma-intersection subsets and explores the boundary between different classes of separation axioms using novel counterexamples. We present several new results concerning separation properties and their behavior under topological operations including quotient maps and product constructions. Our findings provide an enhanced understanding of the separation axiom hierarchy and offer funda-mental insights for studying spaces that lie outside traditional classification schemes. The results have applications in domain theory, functional analysis, and theoretical computer science where such intermediate separation conditions naturally arise.