Difference between revisions of "Urysohn space"
From Encyclopedia of Mathematics
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''space satisfying the Urysohn separation axiom'' | ''space satisfying the Urysohn separation axiom'' | ||
| − | A [[ | + | A [[topological space]] in which any two distinct points have neighbourhoods with disjoint closure. |
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====Comments==== | ====Comments==== | ||
| − | Regular | + | 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">[1]</TD> <TD valign="top"> P.S. Aleksandrov, P. Urysohn, "Mémoire sur les espaces topologiques compacts" , Koninkl. Nederl. Akad. Wetensch. , Amsterdam (1929)</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) pp. 125 (Translated from Russian)</TD></TR> |
| + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Engelking, "General topology" , Heldermann (1989)</TD></TR></table> | ||
Latest revision as of 14:12, 8 April 2023
space satisfying the Urysohn separation axiom
A topological space in which any two distinct points have neighbourhoods with disjoint closure.
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
| [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) |
| [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
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