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T_{YS}

Last-modified: 2013-04-19 (金) 13:27:16

Definition Edit

  • A topological space (X,τ) is is said to be T_{YS} if for all imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20x%2cy%5cin%20X%2c%5c%20x%5cneq%20y%2c%20%5coverline%7b%5c%7bx%5c%7d%7d%5ccap%5coverline%7b%5c%7by%5c%7d%7d%20%5cmbox%7b%20is%20either%20%7d%5cemptyset%20%5c%5d%7d%25.png or {x} or {y}.

Property Edit

  • A topological space (X,τ) is a T_{YS} space iff one of the following conditions holds:
    1. ∀x,y∈X, imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20x%5cneq%20y%20%5c%5d%7d%25.png 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%7bx%5c%7d'%5ccap%5c%7by%5c%7d'=%20%5cemptyset%20%5c%5d%7d%25.png. [3]
    2. ∀x,y∈X, imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20x%5cneq%20y%20%5c%5d%7d%25.png 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%7bx%5c%7d%5ccap%20%5crm%7bD%7d%5c%7by%5c%7d=%20%5cemptyset%20%5c%5d%7d%25.png, where D{x} is the essential derived set of a point x. [3]
  • T_{YS} implies T_Y.
  • T_{YS} = T_0 + R_YS?. [2]

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. Misra, D. N.; Dube, K. K., Some axioms weaker than the R0-axiom., (Serbo-Croatian summary), Glasnik Mat. Ser. III 8(28) (1973), 145-148.
  3. Guia, Josep, Axioms weaker than R0., (Serbo-Croatian summary), Mat. Vesnik 36 (1984), no. 3, 195–205.