Much of the following is based on Counterexamples in Topology by Lynn Arthur Steen and J. Arthur Seebach, Jr.

## Separation Axioms

Let (X,O) be a topological space.

### T0 (Kolmogorov) Spaces

(X,O) is a T0 space if for every pair of points a and b there exists an open set U in O such that at least one of the following statements is true:
1. a lies in U and b does not lie in U.
2. b lies in U and a does not lie in U.

### T1 (Fréchet) Spaces

(X,0) is a T1 space if for every pair of points a and b there exists an open set U such that U contains a but not b. To say that a space is T1 is equivalent to saying that sets consisting of a single point are closed. All T1 spaces are T0. The 'particular point' topology (where the open sets are the sets containing a particular point a) is T0 but not T1.

### T2 (Hausdorff) Spaces

(X,0) is a T2 space if for every pair of points a and b there exist disjoint open sets which separately contain a and b. Some people say in this case that open sets separate points. All T2 spaces are T1. The 'cofinite' topology on an infinite set (where the open sets are those with finite complement) is T1 but not T2.

### T3 (Regular) Spaces

(X,0) is a T3 space if for every point a and closed set B there exist disjoint open sets which separately contain a and B. That is, points and closed sets are separated. Many authors require that T3 spaces also be T0, since with this added condition, they are also T2. The Zariski topology on a vector space (whose closed sets are the intersections of zeroes of polynomials on the coordinates of the vector space) is T2 but not T3.

### T3 1/2 (Completely Regular or Tychonoff) Spaces

(X,0) is a T1/2 space if for every point a and closed set B with a not in B, there exists a continuous function f from X to the interval [0,1] such that f(a) = 0 and f(B) = 1. All T1/2 spaces are T3. Many authors require that T1/2 spaces also be T0. The 'Tychonoff corkscrew' is T3 but not T1/2: I would welcome either a simple description of it, or any other such example.

### T4 (Normal) Spaces

(X,0) is a T4 space if for every pair of closed sets A and B there exist disjoint open sets which separately contain A and B. That is, points and closed sets are separated. Many authors require that T4 spaces also be T1, since with this added condition, they are also T3. Every metric space is T4. An uncountable product of copies of the positive integers (with the discrete topology) is T1/2 but not T4, see Steen and Seebach for an explanation.

### Other Separation Criteria

Steen and Seebach also define T1/2 (or completely Hausdorff), T5, Urysohn, perfectly T4, perfectly normal and semiregular spaces. The interested reader is invited to consult that reference.