Some Properties of R i -Axioms via Multi-Set Topological Spaces

multi-set are sets that are allowed to have repeated members, that is a multi-set MM on a set UU is a count function CMCM from UU to non-negative numbers. This study focuses on introducing and analyzing two new classes of separation axioms named, MM-R0R0 and MM-R1R1 in the context of multi-set topological spaces by utilizing the concepts of distinct MM-singletons and m-closure operator, investigating certain properties and characterizing them with some illustrative examples. Relationships with other MM-separation axioms are explored, and it is demonstrated that MM-R0R0 and MM-R1R1 are special cases of MM-regularity. Furthermore, we show that in the context of compact MM-spaces, MM-R1R1 is equivalent to whole MM-regularity. Finally, the hereditary property of these classes is examined.