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''attainable space''
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A [[Topological space|topological space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092030/t0920302.png" /> in which the closure of any one-point set coincides with itself. This is equivalent to the requirement that the intersection of all neighbourhoods of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092030/t0920303.png" /> is identical with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092030/t0920304.png" /> or that, for any two different points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092030/t0920305.png" />, there exist neighbourhoods <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092030/t0920306.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092030/t0920307.png" /> of them such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092030/t0920308.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092030/t0920309.png" />, i.e. that the [[Separation axiom|separation axiom]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092030/t09203010.png" /> holds.
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Attainability, i.e. the property that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092030/t09203011.png" /> holds, is a hereditary property: Any subspace of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092030/t09203012.png" />-space is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092030/t09203013.png" />-space, and a topology majorizing the topology of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092030/t09203014.png" />-space is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092030/t09203015.png" />-topology. Any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092030/t09203016.png" />-space (cf. [[Hausdorff space|Hausdorff space]]) is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092030/t09203017.png" />-space, but the converse is not true: There exist <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092030/t09203018.png" />-spaces which are not <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092030/t09203019.png" />-spaces. These include, for example, an infinite set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092030/t09203020.png" /> with the topology in which the sets with finite complements are considered to be open.
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''attainable space''
  
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A [[Topological space|topological space]]  $  X $
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in which the closure of any one-point set coincides with itself. This is equivalent to the requirement that the intersection of all neighbourhoods of a point  $  x \in X $
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is identical with  $  x $
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or that, for any two different points  $  x , y \in X $,
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there exist neighbourhoods  $  U _ {x} $
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and  $  U _ {y} $
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of them such that  $  U _ {x} \Nso y $
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and  $  U _ {y} \Nso x $,
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i.e. that the [[Separation axiom|separation axiom]]  $  T _ {1} $
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holds.
  
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Attainability, i.e. the property that  $  T _ {1} $
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holds, is a hereditary property: Any subspace of a  $  T _ {1} $-
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space is a  $  T _ {1} $-
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space, and a topology majorizing the topology of a  $  T _ {1} $-
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space is a  $  T _ {1} $-
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topology. Any  $  T _ {2} $-
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space (cf. [[Hausdorff space|Hausdorff space]]) is  $  T _ {1} $-
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space, but the converse is not true: There exist  $  T _ {1} $-
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spaces which are not  $  T _ {2} $-
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spaces. These include, for example, an infinite set  $  \beta $
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with the topology in which the sets with finite complements are considered to be open.
  
 
====Comments====
 
====Comments====
A very important class of spaces that are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092030/t09203021.png" /> but, as a rule, not <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092030/t09203022.png" /> are the spectra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092030/t09203023.png" /> of rings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092030/t09203024.png" /> with the Zariski topology, cf. [[Affine scheme|Affine scheme]].
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A very important class of spaces that are $  T _ {1} $
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but, as a rule, not $  T _ {2} $
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are the spectra $  \mathop{\rm Spec} ( A) $
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of rings $  A $
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with the Zariski topology, cf. [[Affine scheme|Affine scheme]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Engelking,  "General topology" , Heldermann  (1989)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Engelking,  "General topology" , Heldermann  (1989)</TD></TR></table>

Latest revision as of 08:25, 6 June 2020


attainable space

A topological space $ X $ in which the closure of any one-point set coincides with itself. This is equivalent to the requirement that the intersection of all neighbourhoods of a point $ x \in X $ is identical with $ x $ or that, for any two different points $ x , y \in X $, there exist neighbourhoods $ U _ {x} $ and $ U _ {y} $ of them such that $ U _ {x} \Nso y $ and $ U _ {y} \Nso x $, i.e. that the separation axiom $ T _ {1} $ holds.

Attainability, i.e. the property that $ T _ {1} $ holds, is a hereditary property: Any subspace of a $ T _ {1} $- space is a $ T _ {1} $- space, and a topology majorizing the topology of a $ T _ {1} $- space is a $ T _ {1} $- topology. Any $ T _ {2} $- space (cf. Hausdorff space) is $ T _ {1} $- space, but the converse is not true: There exist $ T _ {1} $- spaces which are not $ T _ {2} $- spaces. These include, for example, an infinite set $ \beta $ with the topology in which the sets with finite complements are considered to be open.

Comments

A very important class of spaces that are $ T _ {1} $ but, as a rule, not $ T _ {2} $ are the spectra $ \mathop{\rm Spec} ( A) $ of rings $ A $ with the Zariski topology, cf. Affine scheme.

References

[a1] R. Engelking, "General topology" , Heldermann (1989)
How to Cite This Entry:
T 1-space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=T_1-space&oldid=18177
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article