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R_0

Last-modified: 2011-07-10 (日) 18:18:23

Definition Edit

Property Edit

  • A topological space (X, τ) is R_0 if it satisfies one of the following equivalent conditions:
    1. imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20x%5cin%20U%5cin%20%5ctau%20%5c%5d%7d%25.png implies imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20%5cmathrm%7bcl%7d%5c%7bx%5c%7d%5csubset%20U%20%5c%5d%7d%25.png.
    2. cl({x})=∩{U | U is an open neighbourhood of x} for each x in X.
    3. cl({x})=cl({y}) or cl({x}) ⋂ cl({y})= imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20%5cemptyset%20%5c%5d%7d%25.png for x, y in X.
    4. The specialization preoreder? on X is symmetric?.
    5. The fixed ultrafilter at x convereges only to the points that x is topologically indistinguishable from.
    6. The Kolmogorov quotient? of X ( which identifies topologically indistinguishable points) is T_1
    7. Every open set is the union of closed sets.
  • A topological space (X,τ) is R_0 iff for any point x in X, cl({x}) ⊆ ker(x).
  • An R_0 space is a w-R_0 space. The converse does not hold.
  • R_0 ⇒ C_0. [4]
  • R_0 ⇒ R_T. [2]
  • R_0 ⇒ R_YS?. [3]
  • R_0 ⇒ R_D. [3]
  • R_0 ⇒ R*_D. [4]
  • R_0 + normal ⇒ regular. [3]

Reference Edit

  1. 忘れた。
  2. Tong, Jing Cheng, On the separation axiom R0. (Serbo-Croatian summary), Glas. Mat. Ser. III 18(38) (1983), no. 1, 149–152.
  3. 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.
  4. 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.