(α,β)-SEMI CONNECTED IN TOPOLOGICAL SPACES

In this paper we used the definition of (α, β)-semi open sets in order to define (α,β)-semi connected spaces in a topological space (X,τ ). Also we study some properties of (α,β)-semi connected spaces and characterize the (α,β)-semi connectedness using ((α, β), (σ, θ)) irresolute maps. Also we can see that this concept generalize the notions of connected, α connected and α semi connected studied before. In this section, we recall some basic definitions and some important results. Definition 2.1. Let (X,τ) be a topological space. We say that α is an ope rator associated to τ ,i fα:P (X)→P (X) satisfies U ⊆ α(U ), for all U ∈ τ. Definition 2.2. Let (X,τ) be a topological space and α:P (X)→P (X) be an operator asso- ciated to a topology τ. A subset A ⊆X is said to be an α-semi open set if there exists U ∈τ such that U ⊆A⊆α(U ). Definition 2.3. Let (X,τ) be a topological space and α,β:P (X)→P (X) be operators asso- ciated to a topology τ on X. We say that a subset A⊆X is an (α,β)-semi open set if for each x∈A, there exists a β-semi open set V such that x ∈ V and α(V ) ⊆ A. The complement of an (α, β)-semi open set is an (α, β)-semi closed set. We observe that when α = β = id, we have A is (α,β)-semi open set ⇔ A is open If β = id and α is arbitrary, then A is (α,β)-semi open set ⇔ A is α-open set