Weak Forms of ?-Open Sets and Decomposition of Separation Axioms

The goal of this work is to recall some types of weak open sets, prove some of its properties and use them to define new kinds of separation axioms. Let us state below: some of our important main theoremsLet (X,σ) and (Y,τ) be two topological spaces satisfy the ω-condition then the map f: (X,σ)⟶(Y,τ) is continuous if and only if it is ω-continuous. ( This result is not true without ω-condition ). Let (X,σ) and (Y,τ) be two topological spaces satisfy the ω-B_α-condition then the map f: (X,σ)⟶(Y,τ) is continuous if and only if it is α-ω-continuous. Let (X,σ) and (Y,τ) be two topological spaces satisfy the ω-B-condition then the map f: (X,σ)⟶(Y,τ) is continuous if and only if it is pre-ω-continuous. Let (X,σ) and (Y,τ) be two door topological spaces and f: (X,σ)⟶(Y,τ) be a map, then f is pre-ω-continuous if and only if it is ω-continuous. And f is β-ω-continuous if and only if it is b-ω-continuous. Let f: X⟶Y be an ω-continuous map from the ω-compact space Xonto a topological space Y. Then Yis ω-compact space. (Similarly for the other types of the weak continuity and compact