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Last-modified: 2011-08-21 (日) 16:11:41

Definition Edit

  • 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 Edit

  • 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 imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20A%5ccap%20U%5cneq%20%5cemptyset%20%5c%5d%7d%25.png, there exists a V ∈ γ-βO(X) such that imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20A%5ccap%20V%5cneq%20%5cemptyset%20%5c%5d%7d%25.png and γ-βcl(V) ⊂ U.
    5. For each nonempty subset A of X and each γ-β-closed subset F of X with imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20A%5ccap%20F=%20%5cemptyset%20%5c%5d%7d%25.png, there exists V,W ∈ γ-βO(X) such that imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20A%5ccap%20V%5cneq%20%5cemptyset%20%5c%5d%7d%25.png, F ⊂ W and imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20W%5ccap%20V=%20%5cemptyset%20%5c%5d%7d%25.png.
    6. For each γ-β -closed set F and imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20x%5cnotin%20F%20%5c%5d%7d%25.png, there exists U ∈ γ-βO(X) and a γ-βg-open set V such that x ∈ U, F ⊂ V and imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20U%5ccap%20V=%20%5cemptyset%20%5c%5d%7d%25.png.
    7. For each A ⊂ X and each γ-β-closed set F with imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20A%5ccap%20F=%20%5cemptyset%20%5c%5d%7d%25.png, there exists U ∈ γ-βO(X) and a γ-βg-open set V such thatimgtex.fcgi?%5bres=100%5d%7b%5c%5b%20A%5ccap%20U%5cneq%20%5cemptyset%20%5c%5d%7d%25.png, F ⊂ V and imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20U%5ccap%20V=%20%5cemptyset%20%5c%5d%7d%25.png.
    8. For each γ-β-closed set F of X, F =∩{γ-βcl(V) : F ⊂ V, V is γ-βg-open}.

Reference Edit

  • 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.