γ-β-open

Definition

• Let (X,τ) be a topological space, an operation γ on τ and A ⊂ X. Then A is called a γ-β-open set if A ⊂ τ_γ-cl(τ_γ-int(τ_γ-cl(A))).

Property

• The concepts of β-open and γ-β-open sets are independent, while in a β-regular? space these concepts are equivalent.
• A β-open set is γ-β-open but the converse may not be true.
• A β-semiopen? set is γ-β-open and it is quite clear that the converse is true when A is γ-closed.
• An arbitrary intersection of γ-β-closed sets is γ-β-closed.

Remark

• See τ_γ-interior and τ_γ-closure.
• The intersection of even two γ-β-open sets may not be γ-β-open.

Reference

• Basu, C. K.; Afsan, B. M. Uzzal; Ghosh, M. K., A class of functions and separation axioms with respect to an operation. (English summary), Hacet. J. Math. Stat. 38 (2009), no. 2, 103-118.