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T_F

Last-modified: 2013-04-19 (金) 13:26:36

Definition Edit

  • A topological space (X,τ) is said to be T_F if given any point x and any finite set F such that imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20x%5cnotin%20F%20%5c%5d%7d%25.png , either {x} is weakly separated from F or F is weakly separated from {x}.

Property Edit

  • A topological space (X,τ) is a T_F space iff one of the following conditions holds:
    1. ∀x, y∈ X, y∈{x}' implies imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20%5crm%7bcl%7d%28%5c%7bx%5c%7d%29%5ccap%5crm%7bker%7d%28x%29%5cneq%20%5crm%7bcl%7d%28%5c%7by%5c%7d%29%5ccap%5crm%7bker%7d%28y%29%20%5c%5d%7d%25.png and imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20%5c%7by%5c%7d'=%5cemptyset%20%5c%5d%7d%25.png. [2]
    2. ∀x, y∈ X, y∈{x}' implies imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20%5crm%7bcl%7d%28%5c%7bx%5c%7d%29%5ccap%5crm%7bker%7d%28x%29%5cneq%20%5crm%7bcl%7d%28%5c%7by%5c%7d%29%5ccap%5crm%7bker%7d%28y%29%20%5c%5d%7d%25.png and imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20%5crm%7bD%7d%5c%7by%5c%7d=%20%5cemptyset%20%5c%5d%7d%25.png, where D{y} is the essential derived set of a point y. [2]
  • T_F ⇒ T_{UD}.

Reference Edit

  1. Aull, Charles E.; Thron, W.J.,Separation axioms between T0 and T1., (English) [J] Nederl. Akad. Wet., Proc., Ser. A 65, 26-37 (1962).
  2. -Guia, Josep, Axioms weaker than R0., (Serbo-Croatian summary), Mat. Vesnik 36 (1984), no. 3, 195–205.