A T₀ space, also known as a Kolmogorov space, is a topological space where for every pair x ≠ y, there is an open set containing one but not the other.

A T₁ space, also known as a Fréchet space, is a topological space where every singleton set is closed. Equivalently, for every pair x ≠ y, there is an open set containing x and not y.

A T₂ space, also known as a Hausdorff space, is one in which for every x ≠ y there exists disjoint open sets around x and y. This is the most widely used of the separation axioms.

A T₃ space, also known as a regular space (although this condition sometimes omits T₂), is one in which for every closed C and x ∉ C, there exist disjoint open sets containing x and C respectively.

A T₄ space, also known as a normal space (although this condition sometimes omits T₂), is one in which for every pair of disjoint closed sets C and D, there exist disjoint open sets containing C and D respectively.