separation axiom (idea)

There is a hierarchy of separation axioms which impose successively stronger conditions on a topological space X, all having to do with the existence of open
neighborhoods separating two fixed points or closed sets in X. In a general topology book they are numbered using the letter T (from the German trennbar, separable, although the concept called separable in English is only peripherally related). All the axioms listed below are inequivalent, although examples that show T₃ ≠ T_3.5 ≠ T₄ are a little hard to construct. (Evandar has thoughtfully cited some below.) From weakest to strongest:

T₀

X is T₀ if, given two distinct points x, y in X, one of them has a neighborhood not containing the other. A T₀-space which is not T₁ is N (the natural numbers) with the topology whose open sets are [n, ∞) for every n in N.

T₁

X is T₁ if, given two distinct points x, y in X, each of them has a neighborhood not containing the other. This is the weakest separation axiom which implies that a one-point set is closed. At least in the classical situation, the Zariski topology on Cⁿ (n-dimensional complex affine space) is a T₁ topology which is not T₂.

T₂

This is the Hausdorff axiom: given two distinct points x, y in X, there are disjoint neighborhoods U of x and V of y. Evandar has supplied an example to show that T₂ ≠ T₃ in his writeup below.

T₃

A space is T₃ or regular if, given x in X and a closed set F not containing x, there are disjoint open sets U containing x and V containing F.

T_3.5

A space is T_3.5 or completely regular if, given x and F as for T₃, not only are x and F separated by open sets, but by a continuous function: there is a continuous map f: X → [0,1] such that f(x) = 0 and f(F) = {1}. Sometimes a T_3.5 space is called a Tychonoff space (various spellings), after the guy who proved that a product of compact spaces is compact.

T₄

A space is T₄ or normal if, given two disjoint closed sets F and G, there are disjoint open sets U containing F and V containing G. Unlike all the previous axioms, this one is not inherited by subspaces of X, and products of normal spaces need not be normal (the Sorgenfrey line is an example of the second phenomenon). Every metrizable space is normal. There is no "T_4.5" analogous to T_3.5; in any normal space, two disjoint closed sets can be separated by a continuous function. This is the content of Urysohn's lemma which is often called the first nontrivial theorem of general topology.