R_0

Definition

• A topological space X is a R_0 space if any two topologically distinguishable points in X can be separated

Property

• A topological space (X, τ) is R_0 if it satisfies one of the following equivalent conditions:
  1. implies .
  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})= 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

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.