The axioms of a topological space do not themselves guarantee sufficient open sets to separate out the space; there are various separation axioms which address this problem, most notably the T_n axioms. A space is: T₀ if, given distinct points, there is an open set which contains one and not the other; T₁ if, given distinct points, each point is contained in an open set and not the other. This is equivalent to points being closed; T₂ if it is a Hausdorff space; T₃ if it is a regular space and T₁; T₄ if it is a normal space and T₁; T₅ if it is a completely normal space and T₁; T₆ if it is a perfectly normal space and T₁.

A Tychonoff space is T₃ but need not be T₄ and so is often referred to as T_3.5. All these separation axioms are strictly nested, that is, if m > n, then a T_m space is T_n and there exists a T_n space which is not T_m. Urysohn’s lemma shows that T₄ spaces are T_3.5. Metric spaces are T₆.