A T_1-space is a topological space fulfilling the T_1-separation axiom: For any two points x, y element X there exists two open sets U and V such that x element U and y not element U, and y element V and x not element V. In the terminology of Alexandroff and Hopf, 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 T_2-spaces.

Banach space | Hausdorff axioms | Hilbert space | separation axioms | T_0-space | T_2-space | T_3-space | T_4-space | topological vector space