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T_0

Last-modified: 2011-12-27 (火) 20:47:08

Definition Edit

Property Edit

  • A topological space (X,τ) is a T_0-space iff one of the following conditions holds:
    1. ∀x in X, the derived set {x}' is a union of closed sets. [2]
    2. ∀x in X, cl({x}) ∩ ker(x) = {x}. [2]
    3. ∀x in X, {x}' = D{x} where D{x} is the essential derived set of x. [2]
    4. ∀x, y in X, y ∈ {x}' implies imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20x%20%5cnotin%20%5c%7by%5c%7d'%20%5c%5d%7d%25.png. [2]
    5. ∀x, y in X, y ∈ {x}' implies imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20%5c%7by%5c%7d'%5csubsetneq%5crm%7bD%7d%5c%7bx%5c%7d%20%5c%5d%7d%25.png. [2]
    6. ∀x, y in X, y ∈ {x}' implies y ∈ D{x}. [2]
    7. ∀x, y in X, imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20x%20%5cneq%20y%20%5cmbox%7b%20implies%20%7d%20%5crm%7bcl%7d%28%5c%7bx%5c%7d%29%5ccap%20%5crm%7bker%7d%28x%29%5cneq%5crm%7bcl%7d%28%5c%7by%5c%7d%29%5ccap%20%5crm%7bker%7d%28y%29%20%5c%5d%7d%25.png. [2]
  • T_0 ⇔ λ-T_1. [4]
  • T_0 ⇒ λ-T_0. [4]
  • T_0 = T_R + R*_0. [3]

Reference Edit

  1. ??
  2. Guia, Josep, Axioms weaker than R0., (Serbo-Croatian summary), Mat. Vesnik 36 (1984), no. 3, 195–205.
  3. Guia, Josep, Essentially T_D and essentially T_UD spaces., Bull. Math. Soc. Sci. Math. R. S. Roumanie (N.S.) 32(80) (1988), no. 3, 227-233.
  4. Caldas Miguel, Jafari Saeid, Navalagi Govindappa, More on λ-closed sets in topological spaces.,Rev. Colombiana Mat. 41 (2007), no. 2, 355-369.