Separation Axioms

A list of five properties of a topological space  expressing how rich the "population" of open sets is. More precisely, each of them tells us how tightly a closed subset can be wrapped in an open set. The measure of tightness is the extent to which this envelope can separate the subset from other subsets. The numbering from 0 to 4 refers to an increasing degree of separation.

0. T0-separation axiom: For any two points , there is an open set  such that  and  or  and .

1. T1-separation axiom: For any two points  there exists two open sets  and  such that  and , and  and .

2. T2-separation axiom: For any two points  there exists two open sets  and  such that , , and .

3. T3-separation axiom:  fulfils  and is regular.

4. T4-separation axiom:  fulfils  and is normal.

Some authors (e.g., Cullen 1968, pp. 113 and 118) interchange axiom  and regularity, and axiom  and normality.

A topological space fulfilling  is called a -space for short. In the terminology of Alexandroff and Hopf (1972), -spaces are also called Kolmogorov spaces, -spaces are Fréchet spaces, -spaces are Hausdorff spaces, -spaces are Vietoris spaces, and -spaces are Tietze spaces. These names can also be referred to the topologies.

A topological space fulfilling one of the axioms also fulfils all preceding axioms, since . None of these implications can be reversed in general. This is possible only under additional assumptions. For example, a regular -space is , and a compact -space is  (McCarty 1967, p. 145). A metric topology is always , whereas the trivial topology on a space with at least two elements is not even . An example of a topology that is  but not  is the one whose open sets are the intervals  of the real line. Given two distinct real numbers , if , then , but . This shows that axiom  is fulfilled. Axiom  is not, since it can be easily shown that  is true iff all singleton sets are closed. For this reason, the Zariski topology of  is . However, it is not , because the intersection of two open sets is always nonempty.

Note that in this context the word axiom is not used in the meaning of "principle" of a theory, which has necessarily to be assumed, but in the meaning of "requirement" contained in a definition, which can be fulfilled or not, depending on the cases.

REFERENCES:

Alexandroff, P. and Hopf, H. Topologie, Vol. 1. New York: Chelsea 1972.

Cullen, H. F. "Separation Axioms." Ch. 3 in Introduction to General Topology. Boston, MA: Heath, pp. 99-140, 1968.

Joshi, K. D. "Separation Axioms." Ch. 7 in Introduction to General Topology. New Delhi, India: Wiley, pp. 159-188, 1983.

McCarty, G. Topology, an Introduction with Application to Topological Groups. New York: McGraw-Hill, 1967.

Willard, S. "The Separation Axioms." §13 in General Topology. Reading, MA: Addison-Wesley, pp. 85-92, 1970.