γ-β-regular

Definition

• A topological space (X,τ) with an operation γ on τ is called γ-β-regular if for each
  γ-β-closed set F of X not containing x, there exist disjoint
  γ-β-open sets U and V such that x ∈ U and F ⊂ V .

Property

• The following are equivalent for a topological space (X,τ) with an operation γ on τ :
  1. X is γ-β-regular.
  2. For each x ∈ X and each U ∈ γ-βO(X, x), there exists a V ∈ γ-βO(X, x) such that x ∈ V ⊂ γ-βcl(V) ⊂ U.
  3. For each γ-β-closed set F of X, ∩{γ-βcl(V) : F ⊂ V, V ∈ γ-βO(X)} = F.
  4. For each A subset of X and each U ∈ γ-βO(X) with , there exists a V ∈ γ-βO(X) such that and γ-βcl(V) ⊂ U.
  5. For each nonempty subset A of X and each γ-β-closed subset F of X with , there exists V,W ∈ γ-βO(X) such that , F ⊂ W and .
  6. For each γ-β-closed set F and , there exists U ∈ γ-βO(X) and a γ-βg-open set V such that x ∈ U, F ⊂ V and .
  7. For each A ⊂ X and each γ-β-closed set F with , there exists U ∈ γ-βO(X) and a γ-βg-open set V such that , F ⊂ V and .
  8. For each γ-β-closed set F of X, F =∩{γ-βcl(V) : F ⊂ V, V is γ-βg-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.