Description
Every T1 space is T0 Space | Theorem | separation axioms | Topology
T0 Space :
A topological space fulfilling the T0-separation axiom: For any two points x,y in X, there is an open set U such that x in U and y not in U or y in U and x not in U. Alexandroff and Hopf (1972) call T_0-spaces Kolmogorov spaces.
T1 Space:
A T_1-space is a topological space fulfilling the T1-separation axiom: For any two points x,y in X there exists two open sets U and V such that x in U and y not in U, and y in V and x not in V. In the terminology of Alexandroff and Hopf (1972), T_1-spaces are known as Fréchet spaces (though this is confusing and nonstandard).
The standard example of a T_1-space is the set of integers with the topology of open sets being those with finite complements. It is closed under finite intersection and arbitrary union so is a topology. Any integer’s complement is an open set, so given two integers and using their complement as open sets, it follows that the T_1 definition is satisfied. Some T_1-spaces are not T2-spaces.