T_0
Last-modified: 2011-12-27 (火) 20:47:08
Definition
Property
- A topological space (X,τ) is a T_0-space iff one of the following conditions holds:
- ∀x in X, the derived set {x}' is a union of closed sets. [2]
- ∀x in X, cl({x}) ∩ ker(x) = {x}. [2]
- ∀x in X, {x}' = D{x} where D{x} is the essential derived set of x. [2]
- ∀x, y in X, y ∈ {x}' implies
. [2]
- ∀x, y in X, y ∈ {x}' implies
. [2]
- ∀x, y in X, y ∈ {x}' implies y ∈ D{x}. [2]
- ∀x, y in X,
. [2]
- T_0 ⇔ λ-T_1. [4]
- T_0 ⇒ λ-T_0. [4]
- T_0 = T_R + R*_0. [3]
Reference
- ??
- Guia, Josep, Axioms weaker than R0., (Serbo-Croatian summary), Mat. Vesnik 36 (1984), no. 3, 195–205.
- 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.
- Caldas Miguel, Jafari Saeid, Navalagi Govindappa, More on λ-closed sets in topological spaces.,Rev. Colombiana Mat. 41 (2007), no. 2, 355-369.