A NOTE ON TOPOLOGIES GENERATED BY $m$-STRUCTURES AND $!$-TOPOLOGIES

Abstract

Let $ $(resp. SO$(X; )$) be the family of all $$-open (resp. semi-open) 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 (mX)$ and$T (!mX)$ for $m$-structrues $mX$ and $!mX$ defined in $[11]$, respectively, and showthat $!T$ (mX) $ T$ $(!mX)$. Furthermore, in [2], a topology $M$ is constructedby using an M-space $(X;M)$ with an ideal I. In this note, we define $!M$-opensets on $(X;M)$ and show that the family $!M$of all $!M$-open sets is a topologyfor X and $!(M) = (!M) = (!M)$: