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γ-βT_2

Last-modified: 2011-08-20 (土) 12:51:27

Definition Edit

  • A topological space (X,τ) with an operation γ on τ is called γ-βT_2 if and only if for each pair of distinct points x, y in X, there exists two disjoint γ-β-open sets U and V containing x and y respectively.

Property Edit

  • The following are equivalent for a topological space (X,τ) with an operation γ on τ :
    1. X is γ-βT_2.
    2. Let x ∈ X. For each imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20y%5cneq%20x%20%5c%5d%7d%25.png, there exists a γ-β-open set U containing x such that imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20y%5cnotin%5cgamma%5c%2c%5cmbox%7b-%7d%5c%2c%5cbeta%5c%2ccl%28U%29%20%5c%5d%7d%25.png.
    3. For each x ∈ X, ∩{γ-βcl(U) : x ∈ U ∈ γ-βO(X)} = {x}.
  • γ-βT_2 ⇒ γ-βT_1.

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.