Asian Journal of Pure and Applied Mathematics

Background: Separation axioms in ideal topological spaces provide a framework for examining how points and subsets may be distinguished using ideal-based open sets. However, the relationships among separation axioms weaker than the R−I −R₀ axiom require further systematic clarification.

Aims: This paper aims to study selected properties of R − I − R1 spaces and to introduce and examine new separation axioms weaker than R − I − R₀ in ideal topological spaces.

Study Design: Theoretical study.

Place of the Study: Department of Mathematics, St. Joseph’s College (Autonomous), Devagiri, Calicut-673008, India.

Methodology: The study uses logical deduction from established concepts in topology and ideal topological spaces. It recalls preliminaries on ideals, local functions, R − I−open sets, R − I−closures, R − I−neighbourhoods and R − I−kernels, and then derives definitions, characterisations and implication results within this framework.

Results: The paper characterises R − I−regular spaces and R − I − R1 spaces through equality conditions involving R−I−closure and R−I −θ−closure. It introduces the classes R−I −R_S, R−I −RD, R−I −R_T , weakly R − I − R₀ and weakly R − I − C₀ spaces. The results show that every R − I − R₀ space is weakly R−I −R₀ and R−I −R_T . It is also shown that every R−I −RT space is both R−I −R_D and R−I −R_S. Further characterisations are obtained for weakly R−I −R₀ and weakly R−I −C0 spaces. A finite example demonstrates that the converse implications from weakly R − I − R0 and R − I − RT to R − I − R0 do not hold in general.

Conclusion: The study clarifies the hierarchy of selected weak separation axioms in ideal topological spaces and positions the newly introduced classes relative to R − I − R0 and R − I − R₁ spaces.