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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.
### T_{0} (Kolmogorov) Spaces

(X,O) is a T_{0} 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:
*a* lies in U and *b* does not lie in U.
*b* lies in U and *a* does not lie in U.

### T_{1} (Fréchet) Spaces

(X,0) is a T_{1} 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 T_{1} is equivalent to saying that sets
consisting of a single point are closed.
All T_{1} spaces are T_{0}.
The 'particular point' topology (where the open sets are the sets
containing a particular point *a*) is T_{0} but not T_{1}.
### T_{2} (Hausdorff) Spaces

(X,0) is a T_{2} 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 T_{2} spaces are T_{1}.
The 'cofinite' topology on an infinite set (where the open sets are those
with finite complement) is T_{1} but not T_{2}.
### T_{3} (Regular) Spaces

(X,0) is a T_{3} 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 T_{3} spaces also be T_{0},
since with this added condition, they are also T_{2}.
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 T_{2} but not T_{3}.
### T_{3 1/2}
(Completely Regular or Tychonoff) Spaces

(X,0) is a T_{3 1/2} 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 T_{3 1/2} spaces are T_{3}.
Many authors require that T_{3 1/2} spaces
also be T_{0}.
The 'Tychonoff corkscrew' is T_{3} but not
T_{3 1/2}: I would welcome either a simple
description of it, or any other such example.
### T_{4} (Normal) Spaces

(X,0) is a T_{4} 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 T_{4} spaces also be T_{1},
since with this added condition, they are also T_{3}.
Every metric space is T_{4}.
An uncountable product of copies of the positive integers (with the
discrete topology) is T_{3 1/2} but not
T_{4}, see Steen and Seebach for an explanation.
### Other Separation Criteria

Steen and Seebach also define T_{2 1/2}
(or completely Hausdorff),
T_{5},
Urysohn,
perfectly T_{4},
perfectly normal
and semiregular spaces.
The interested reader is
invited to consult that reference.