γ-β-normal

Definition

• A topological space (X,τ) with an operation γ on τ is said to be γ-β-
  normal if for any pair of disjoint γ-β-closed sets A, B of X, there exist disjoint γ-β-open sets U and V such that A ⊂ U and B ⊂ V .

Property

For a topological space (X,τ) with an operation γ on τ, the following are equivalent:

1. X is γ-β-normal.
2. For each pair of disjoint γ-β-closed sets A, B of X, there exist disjoint γ-βg-open
   sets U and V such that A ⊂ U and B ⊂ V .
3. For each γ-β-closed A and any γ-β-open set V containing A, there exists a　γ-βg-open set U such that A ⊂ U ⊂ γ-βcl(U) ⊂ V .
4. For each γ-β-closed set A and any γ-βg-open set B containing A, there exists a γ-βg-open set U such that A ⊂ U ⊂ γ-βcl(U) ⊂ γ-βint(B).
5. For each γ-β-closed set A and any γ-βg-open set B containing A, there exists a γ-β-open set G such that A ⊂ G ⊂ γ-βcl(G) ⊂ γ-βint(B).
6. For each γ-βg-closed set A and any γ-β-open set B containing A, there exists a γ-β-open set U such that γ-βcl(A) ⊂ U ⊂ γ-βcl(U) ⊂ B.
7. For each γ-βg-closed set A and any γ-β-open set B containing A, there exists a γ-βg-open set G such that γ-βcl(A) ⊂ G ⊂ γ-βcl (G) ⊂ B.

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.