Difference between revisions of "Separation Axioms"

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{{Under Construction}}
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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>
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== Overview ==
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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>
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{| class="wikitable" style="text-align:center; margin: auto; width: 80%;"
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! scope="col" | Axiom
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! scope="col" | Associated Space
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! scope="col" | Description
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|-
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| $T_0$
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| [[Kolmogorov Space]]
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|
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|-
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| $T_1$
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| [[Fréchet Space]]
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| 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 $y \in O_x$.
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|-
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| $T_2$
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| [[Hausdorff space|Hausdorff Space]]
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| Every pair of disjoint points in a Hausdorff space has a pair of disjoint [[Neighborhood|neighborhood]]s.
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|-
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| $T_3$
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| [[Regular Space|Regular Space]]
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|
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|-
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| $T_4$
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| [[Normal Space|Normal Space]]
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|
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|-
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| $T_5$
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| [[Completely Normal Space]]
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|
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|}
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== Further Reading ==
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* [[Countability Axioms]]
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* [[Limit point|Limit Point]]
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* [[Normal Space]]
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* [[Regular Space]]
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== See Also ==
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* [[Kolmogorov Space]]
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* [[Fréchet Space]]
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* [[Hausdorff space|Hausdorff Space]]
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== References ==
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<references group="references" />
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[[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

Further Reading

See Also

References

  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.