# 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]}

## Overview

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 |