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. | + | |
| + | 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|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. | 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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====References==== | ====References==== | ||
| − | <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> | + | <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> | ||
| + | <TR><TD valign="top">[a3]</TD> <TD valign="top"> W.Franz, "General topology" , Harrap (1967) p. 100</TD></TR> | ||
| + | </table> | ||
Revision as of 18:23, 29 September 2013
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 (cf. Normal space), or (an addition by A.N. Tikhonov) if and only if it is regular.
Comments
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
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