Namespaces
Variants
Actions

Difference between revisions of "Urysohn metrization theorem"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (better)
m (better)
 
(3 intermediate revisions by 2 users not shown)
Line 1: Line 1:
 +
{{TEX|done}}
 +
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]].
  
A [[compact space|compact]] or [[countably-compact space|countably compact]] [[Hausdorff space|Hausdorff space]] is metrizable if and only if it has a [[countable base]]: indeed, it is [[homeomorphism|homeomorphic]] to a subset of the [[Hilbert cube]].
+
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.
 
 
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.
 
 
 
 
 
====Comments====
 
 
 
  
 
====References====
 
====References====
Line 14: Line 10:
 
<TR><TD valign="top">[a3]</TD> <TD valign="top"> W.Franz,  "General topology" , Harrap  (1967)  p. 100</TD></TR>
 
<TR><TD valign="top">[a3]</TD> <TD valign="top"> W.Franz,  "General topology" , Harrap  (1967)  p. 100</TD></TR>
 
</table>
 
</table>
 +
 +
[[Category:General topology]]

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=30580
This article was adapted from an original article by P.S. Aleksandrov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article