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Difference between revisions of "Urysohn metrization theorem"

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A compact or countably compact [[Hausdorff space|Hausdorff space]] is metrizable if and only if it has a countable base.
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A [[compact space|compact]] or [[countably-compact space|countably compact]] [[Hausdorff space]] is [[Metrizable space|metrizable]] if and only if it has a [[countable base]]: indeed, it is [[homeomorphism|homeomorphic]] to a subset of the [[Hilbert cube]].
  
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A [[Topological space|topological space]] with a countable base is metrizable if and only if it is normal (cf. [[Normal space|Normal space]]), or (an addition by A.N. Tikhonov) if and only if it is regular.
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A [[topological space]] with a countable base is [[Metrizable space|metrizable]] if and only if it is [[Normal space|normal]], or (an addition by A.N. Tikhonov) if and only if it is regular.
  
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====References====
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  A.V. Arkhangel'skii,  V.I. Ponomarev,  "Fundamentals of general topology: problems and exercises" , Reidel  (1984)  pp. Chapt. 5  (Translated from Russian)</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  J.L. Kelley,  "General topology" , v. Nostrand  (1955)  pp. 125; 127</TD></TR>
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<TR><TD valign="top">[a3]</TD> <TD valign="top"> W.Franz,  "General topology" , Harrap  (1967)  p. 100</TD></TR>
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</table>
  
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[[Category:General topology]]
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====Comments====
 
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====References====
 
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.V. Arkhangel'skii,  V.I. Ponomarev,  "Fundamentals of general topology: problems and exercises" , Reidel  (1984)  pp. Chapt. 5  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.L. Kelley,  "General topology" , v. Nostrand  (1955)  pp. 125; 127</TD></TR></table>
 

Latest revision as of 19:59, 15 October 2014

A compact or countably compact Hausdorff space is metrizable if and only if it has a countable base: indeed, it is homeomorphic to a subset of the Hilbert cube.

A topological space with a countable base is metrizable if and only if it is normal, or (an addition by A.N. Tikhonov) if and only if it is regular.

References

[a1] A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) pp. Chapt. 5 (Translated from Russian)
[a2] J.L. Kelley, "General topology" , v. Nostrand (1955) pp. 125; 127
[a3] W.Franz, "General topology" , Harrap (1967) p. 100
How to Cite This Entry:
Urysohn metrization theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Urysohn_metrization_theorem&oldid=11449
This article was adapted from an original article by P.S. Aleksandrov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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