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

Let τα (resp. SO(X, τ)) be the family of all α-open (resp. semiopen) sets in a topological space (X, τ). The topology τα is constructed in [10] as follows: τα = T (SO(X)) = {U ⊂ X : U ∩ S ∈ SO(X, τ) for every S ∈ SO(X, τ)}. By the same method, we construct topologies T (mX) and T (ωmX) form-structruesmX and ωmX defined in [11], respectively, and show that ωT (mX) ⊂ T (ωmX). Furthermore, in [2], a topologyM∗ is constructed by using an M -space (X,M) with an ideal I. In this note, we define ωM -open sets on (X,M) and show that the family ωM of all ωM -open sets is a topology for X and ω(M∗) = (ωM)∗ = (ωM)∗.