A NOTE ON SEPARATION AXIOMS IN SOFT TOPOLOGICAL SPACE

The classification of topological spaces via separation axioms constitutes a cornerstone of general topology, providing the structural rigor necessary to distinguish distinct points and closed sets through open neighborhoods.In the emerging field of soft set theory, originally proposed by Molodtsov as a mathematical tool for dealing with uncertainty and vagueness, the translation of these classical axioms into Soft Topological Spaces (STS) has engendered a complex and often divergent theoretical landscape.This research report offers an exhaustive examination of separation axioms-specifically T0, T1, T2 (Hausdorff), Regularity (T3), and Normality (T4)within the context of soft topology.We critically analyze the fundamental bifurcation in the literature concerning the definition of the basic "point" in a soft space: the "ordinary point" approach initiated by Shabir and Naz versus the "soft point" approach developed by Zorlutuna, Hussain, and Ahmad.Through a meticulous synthesis of foundational papers and recent advancements from the 2020-2024 period, this report elucidates the structural anomalies that arise when classical intuitions are applied to parameterized soft structures.We investigate the hierarchy of axioms, demonstrating via constructed counterexamples that classical implications such as T2 implies T1 and Regularity Normality are contingent upon specific definitions of "disjointness" (Type-I vs. Type-II) and membership relations (Total vs. Partial).Furthermore, we explore advanced generalizations including pp-soft and tt-soft separation axioms, establishing a comprehensive comparative framework.The analysis confirms that while soft topology inherits the spirit of classical separation, the introduction of the parameter set E imposes a dimensional constraint that necessitates refined, non-trivial definitions to preserve topological invariance and hereditary properties.