A note on topologies generated by m-structures and ꙍ-topologies

Let(resp.SO(X; )) be the family of all -open (resp.semiopen) sets in a topological space (X; ).The topology is constructed in [10] as follows: = T (SO(X)) = fU X : U \ S 2 SO(X; ) for every S 2 SO(X; )g.By the same method, we construct topologies T (m X ) and T (!m X ) for m-structrues m X and !m X de…ned in [11], respectively, and show that !T (m X ) T (!m X ).Furthermore, in [2], a topology M is constructed by using an M -space (X; M) with an ideal I.In this note, we de…ne !M -open sets on (X; M) and show that the family !M of all !M -open sets is a topology for X and !(M ) = (!M) = (!M) :