Separation Axioms

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The Axioms of Separation are additional conditions to specialize the notion of a topological space. Arbitrary topological spaces are too general an object for many problems of analysis and it is therefore useful to impose extra conditions on a topological space.[1]


In addition the the listed separation axioms, completely regular spaces are sometimes referred to as $T_{3\frac{1}{2}}$ spaces, since they technically lie somewhere between regular and normal spaces.[2]

Axiom Associated Space Description
$T_0$ Kolmogorov Space
$T_1$ Fréchet Space For each pair of distinct points $x$ and $y$, their respective neighborhoods $O_x$ and $O_y$ contain one another, i.e. $x \in O_y$ and $y \in O_x$.
$T_2$ Hausdorff Space Every pair of disjoint points in a Hausdorff space has a pair of disjoint neighborhoods.
$T_3$ Regular Space
$T_4$ Normal Space
$T_5$ Completely Normal Space

Further Reading

See Also


  1. Kolmogorov, A. N., and S. V. Fomin. Introductory Real Analysis. Rev. English ed. New York: Dover Publications, 1975. pg. 78-79.
  2. Munkres, James R. Topology, 2015. pg. 209.