Somewhat continuous functions

Abstract

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SOMEWHAT $e$-$\mathcal{I}$-CONTINUOUS AND SOMEWHAT $e$-$\mathcal{I}$-OPEN FUNCTIONS VIA IDEALS

“…Let X be a set and τ and σ be topologies for X. Then τ is said to be weakly equivalent to σ [8] provided that if U ∈ τ and U = ∅, then there is an open set V in (X, σ) such that V = ∅ and V ⊂ U and if U ∈ σ and U = ∅, then there is an open set V in (X, τ ) such that V = ∅ and V ⊂ U . Definition 23.…”

SOMEWHAT r-CONTINUOUS AND SOMEWHAT r-OPEN FUNCTIONS

“…Definition 1. A function f : X → Y is said to be somewhat continuous [4] if for U ∈ σ and f −1 (U ) = φ, there exists an open set V in X such that V = φ and V⊂ f −1 (U ). Definition 2.…”

“…Definition 80. If X is a set and τ and σ are topologies for X then τ is said to be weakly equivalent [4] to σ provided if U∈ τ and U = φ, then there is an open set V in (X, σ) such that V = φ and V⊂ U and if U in σ and U = φ then there is an open set V in (X, τ ) such that V = φ and V ⊂ U. Definition 81.…”

“…Gentre and Hoyle [4] studied the concept of somewhat continuous functions and somewhat open functions. In this paper a new type of somewhat continuous and open functions namely somewhat r-continuous and somewhat r-open func-tions is introduced.…”

“…Somewhat r-Open Functions Definition 85. A function f : (X, τ ) → (Y, σ) is said to be somewhat open[4] provided for U ∈ τ and U = φ there exists an open set V in Y such that V = φ and V ⊂ f (U ).Definition 86. A function f : (X, τ ) → (Y, σ) is said to be somewhat r-open provided for U ∈ τ and U = φ there exists a regular open set V in Y such that V = φ and V ⊂ f (U ).…”