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Difference between revisions of "Urysohn space"

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''space satisfying the Urysohn separation axiom''
 
''space satisfying the Urysohn separation axiom''
  
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====Comments====
 
====Comments====
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Regular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095900/u0959001.png" />-spaces (cf. [[Regular space|Regular space]]; [[Separation axiom|Separation axiom]]) are Urysohn, and Urysohn spaces are Hausdorff (cf. [[Hausdorff space|Hausdorff space]]). Neither implication is reversible.
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Regular $T_1$-spaces (cf. [[Regular space|Regular space]]; [[Separation axiom|Separation axiom]]) are Urysohn, and Urysohn spaces are Hausdorff (cf. [[Hausdorff space|Hausdorff space]]). Neither implication is reversible.
  
 
====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>

Revision as of 07:49, 12 August 2014

space satisfying the Urysohn separation axiom

A topological space in which any two distinct points have neighbourhoods with disjoint closure.

References

[1] P.S. Aleksandrov, P. Urysohn, "Mémoire sur les espaces topologiques compacts" , Koninkl. Nederl. Akad. Wetensch. , Amsterdam (1929)
[2] A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) pp. 125 (Translated from Russian)


Comments

Regular $T_1$-spaces (cf. Regular space; Separation axiom) are Urysohn, and Urysohn spaces are Hausdorff (cf. Hausdorff space). Neither implication is reversible.

References

[a1] R. Engelking, "General topology" , Heldermann (1989)
How to Cite This Entry:
Urysohn space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Urysohn_space&oldid=18186
This article was adapted from an original article by B.A. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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