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T_1

Last-modified: 2013-04-19 (金) 13:03:31

Definition Edit

Property Edit

  • A topological space (X,τ) is a T_1 space iff one of the following conditions holds:
    1. ∀x∈X, {x}' is empty. [5]
    2. ∀x∈X, cl({x})={x}. [5]
    3. ∀x,y∈X, imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20x%5cneq%20y%20%5cmbox%7b%20implies%20%7d%20%5crm%7bcl%7d%28%5c%7bx%5c%7d%29%5ccap%5crm%7bcl%7d%28%5c%7by%5c%7d%29=%5cemptyset%20%5c%5d%7d%25.png. [5]
    4. Every subset of X is a Λ-set. [7]
  • T_1 = T_0 + R_0. [2]
  • T_1 = T_R + R_0. [6]
  • T_1 = T_0 + C_0. [6]
  • T_1 ⇒ nearly T_1. [3]
  • T_1 ⇒ T_{EF}. [4]
  • T_1 ⇒ λ-T_1. [8]

Reference Edit

  1. ???
  2. Dorsett C., Images and hyperspaces of s-essentially T_1 and s-essentially T_2 spaces and semitopological properties. (Serbo-Croatian summary) Glas. Mat. Ser. III 21(41) (1986), no. 2, 415–422.
  3. Przemski M., Nearly T_i-continuous functions and some separation axioms. (Serbo-Croatian summary), Glas. Mat. Ser. III 21(41) (1986), no. 2, 431–435.
  4. McSherry D. M. G. , On separation axioms weaker than T_1. , Proc. Roy. Irish Acad. Sect. A 74 (1974), 115–118.
  5. Guia Josep, Axioms weaker than R0., (Serbo-Croatian summary), Mat. Vesnik 36 (1984), no. 3, 195-205.
  6. 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.
  7. Arenas Francisco G., Dontchev Julian, Ganster Maximilian, On λ-sets and the dual of generalized continuity., Questions Answers Gen. Topology 15 (1997), no. 1, 3-13.
  8. Caldas Miguel, Jafari Saeid, Navalagi Govindappa, More on λ-closed sets in topological spaces.,Rev. Colombiana Mat. 41 (2007), no. 2, 355-369.