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Difference between revisions of "Hausdorff space"

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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046730/h0467302.png" />-space''
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$T_2$-''space''
  
A topological space in which any two (distinct) points are separated by disjoint neighbourhoods (see [[Hausdorff axiom|Hausdorff axiom]]). Hausdorff spaces need not be regular nor a fortiori completely regular, even when they consist only of a countable set of points or have a countable base. They were first considered by F. Hausdorff in 1914, see [[#References|[1]]].
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A topological space in which any two (distinct) points are separated by disjoint [[neighbourhood]]s (see [[Hausdorff axiom|Hausdorff axiom]]). Hausdorff spaces need not be [[regular space]]s nor a fortiori [[completely regular space]]s, even when they consist only of a countable set of points or have a countable base (cf. [[Second axiom of countability]]). They were first considered by F. Hausdorff in 1914, see [[#References|[1]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  F. Hausdorff,  "Set theory" , Chelsea, reprint  (1978)  (Translated from German)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.V. Arkhangel'skii,  V.I. Ponomarev,  "Fundamentals of general topology: problems and exercises" , Reidel  (1984)  (Translated from Russian)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  F. Hausdorff,  "Set theory" , Chelsea, reprint  (1978)  (Translated from German)</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  A.V. Arkhangel'skii,  V.I. Ponomarev,  "Fundamentals of general topology: problems and exercises" , Reidel  (1984)  (Translated from Russian)</TD></TR>
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</table>

Revision as of 16:12, 29 September 2013

$T_2$-space

A topological space in which any two (distinct) points are separated by disjoint neighbourhoods (see Hausdorff axiom). Hausdorff spaces need not be regular spaces nor a fortiori completely regular spaces, even when they consist only of a countable set of points or have a countable base (cf. Second axiom of countability). They were first considered by F. Hausdorff in 1914, see [1].

References

[1] F. Hausdorff, "Set theory" , Chelsea, reprint (1978) (Translated from German)
[2] A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian)
How to Cite This Entry:
Hausdorff space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hausdorff_space&oldid=18077
This article was adapted from an original article by A.V. Arkhangel'skii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article