On decompositions of continuity and some weaker forms of continuity via idealization

Abstract

We give decompositions of continuity and some weaker forms of continuity via idealization using the concepts of AI-sets, αAI-sets, B1I -sets, B2I -sets, B3I -sets, αCI -sets and W LCI -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.…”

“…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. (4) where is -open and is -dense, (5) where is open and is -dense.…”

On subsets of ideal topological spaces

“…A subset A of an ideal space (X, τ, I) is said to be an αB I −set (α I N 3 −set [1]) if A = U ∩V where U ∈ αIO(X) and V is a t−I−set. A subset A of an ideal space (X, τ, I) is said to be an αC I −set [4]…”