Some new sets and decompositions of A I-R-continuity, α-I-continuity, continuity via idealization

Abstract

The concepts of I-R closed set,-set and α I N 5 -set are introduced via idealization. New decompositions of some weaker forms of continuity are obtained by using these sets.

Applications of Codense and Compatible Ideals

“…Then for , the following properties are equivalent: (1) is an open set, (2) is --open and a locally closed set, (3) is pre --open and a locally closed, (4) is semi --open and a locally closed set, (5) is --open and a locally closed set. Proof.…”

“…where is codense, the following properties are equivalent: (1) is -submaximal, (2) (1) is strongly -open, (2) is semi -open, (3) is pre -open, (4) is -open, (5) is open. Proof.…”

“…A subset of an ideal space is said to be a -set [4] ( resp. -set [4] ( -set [1] ) if where is open ( resp.…”

“…Clearly . A subset of an ideal space is said to be an -set [1] if where is open and is -closed. We will denote the family of all -sets of an ideal space by A subset of an ideal space is said to be -closed set [36] if A subset of an ideal space is said to be an -set [4] (resp.…”

“…An ideal on a topological space as a non-empty collection of subsets of satisfying the following two conditions: (1) If and , then ; (2) If and , then . An ideal topological space [7] is a topological space with an ideal on , and is denoted by An ideal in an ideal topological space is called a codense ideal [8] if .…”