T_1-Space

A -space is a topological space fulfilling the T1-separation axiom: For any two points  there exists two open sets  and  such that  and , and  and . In the terminology of Alexandroff and Hopf (1972), -spaces are known as Fréchet spaces (though this is confusing and nonstandard).

The standard example of a -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  definition is satisfied. Some -spaces are not T2-spaces.

REFERENCES:

Alexandroff, P. and Hopf, H. Topologie, Vol. 1. New York: Chelsea, 1972.