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Separation Axioms


A list of five properties of a topological space X 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 x,y in X, there is an open set U such that x in U and y not in U or y in U and x not in U.

1. T1-separation axiom: For any two points x,y in X there exists two open sets U and V such that x in U and y not in U, and y in V and x not in V.

2. T2-separation axiom: For any two points x,y in X there exists two open sets U and V such that x in U, y in V, and U intersection V=emptyset.

3. T3-separation axiom: X fulfils T_1 and is regular.

4. T4-separation axiom: X fulfils T_1 and is normal.

Some authors (e.g., Cullen 1968, pp. 113 and 118) interchange axiom T_3 and regularity, and axiom T_4 and normality.

A topological space fulfilling T_i is called a T_i-space for short. In the terminology of Alexandroff and Hopf (1972), T_0-spaces are also called Kolmogorov spaces, T_1-spaces are Fréchet spaces, T_2-spaces are Hausdorff spaces, T_3-spaces are Vietoris spaces, and T_4-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 T_4=>T_3=>T_2=>T_1=>T_0. None of these implications can be reversed in general. This is possible only under additional assumptions. For example, a regular T_1-space is T_2, and a compact T_2-space is T_3 (McCarty 1967, p. 145). A metric topology is always T_4, whereas the trivial topology on a space with at least two elements is not even T_0. An example of a topology that is T_0 but not T_1 is the one whose open sets are the intervals (a,+infty) of the real line. Given two distinct real numbers x,y, if x<y, then y in (x,+infty), but x not in (x,+infty). This shows that axiom T_0 is fulfilled. Axiom T_1 is not, since it can be easily shown that T_1 is true iff all singleton sets are closed. For this reason, the Zariski topology of R^n is T_1. However, it is not T_2, 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.


See also

T0-Separation Axiom, T0-Space, T1-Separation Axiom, T1-Space, T2-Separation Axiom, T2-Space, T3-Separation Axiom, T3-Space, T4-Separation Axiom, T4-Space

This entry contributed by Margherita Barile

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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.

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Separation Axioms

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Barile, Margherita. "Separation Axioms." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/SeparationAxioms.html

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