# Difference between revisions of "Separation Axioms"

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− | {{ | + | The ''Axioms of Separation'' are additional conditions to specialize the notion of a [[Topological space|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.<ref>Kolmogorov, A. N., and S. V. Fomin. Introductory Real Analysis. Rev. English ed. New York: Dover Publications, 1975. pg. 78-79.</ref> |

+ | |||

+ | == Overview == | ||

+ | In addition the the listed separation axioms, [[Completely Regular Space|completely regular space]]s are sometimes referred to as $T_{3\frac{1}{2}}$ [[Topological space|space]]s, since they technically lie somewhere between regular and normal spaces.<ref>Munkres, James R. Topology, 2015. pg. 209.</ref> | ||

+ | |||

+ | {| class="wikitable" style="text-align:center; margin: auto; width: 80%;" | ||

+ | ! scope="col" | Axiom | ||

+ | ! scope="col" | Associated Space | ||

+ | ! scope="col" | 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|Hausdorff Space]] | ||

+ | | Every pair of disjoint points in a Hausdorff space has a pair of disjoint [[Neighborhood|neighborhood]]s. | ||

+ | |- | ||

+ | | $T_3$ | ||

+ | | [[Regular Space|Regular Space]] | ||

+ | | | ||

+ | |- | ||

+ | | $T_4$ | ||

+ | | [[Normal Space|Normal Space]] | ||

+ | | | ||

+ | |- | ||

+ | | $T_5$ | ||

+ | | [[Completely Normal Space]] | ||

+ | | | ||

+ | |} | ||

+ | |||

+ | == Further Reading == | ||

+ | * [[Countability Axioms]] | ||

+ | * [[Limit point|Limit Point]] | ||

+ | * [[Normal Space]] | ||

+ | * [[Regular Space]] | ||

+ | |||

+ | == See Also == | ||

+ | * [[Kolmogorov Space]] | ||

+ | * [[Fréchet Space]] | ||

+ | * [[Hausdorff space|Hausdorff Space]] | ||

+ | |||

+ | == References == | ||

+ | <references group="references" /> | ||

+ | |||

+ | [[Category:General Topology]] |

## Latest revision as of 06:17, 1 December 2018

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 |