Difference between revisions of "Isolated point"
From Encyclopedia of Mathematics
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A point $a\in A$ such that the intersection of some [[Neighbourhood|neighbourhood]] of $a$ with $A$ consists of the point $a$ alone. | A point $a\in A$ such that the intersection of some [[Neighbourhood|neighbourhood]] of $a$ with $A$ consists of the point $a$ alone. | ||
| − | A subset $A$ with no isolated points is ''dense- | + | A subset $A$ with no isolated points is ''dense-in-itself''; a closed dense-in-itself subset is a ''[[perfect set]]''. |
====References==== | ====References==== | ||
<table> | <table> | ||
| − | <TR><TD valign="top">[1]</TD> <TD valign="top"> Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1978). ''Counterexamples in Topology'' (second edition). Berlin, New York: Springer-Verlag. ISBN 978-0-486-68735-3 | + | <TR><TD valign="top">[1]</TD> <TD valign="top"> Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1978). ''Counterexamples in Topology'' (second edition). Berlin, New York: Springer-Verlag. {{ISBN|978-0-486-68735-3}} {{MR|507446}} {{ZBL|0386.54001}}</TD></TR> |
</table> | </table> | ||
Latest revision as of 08:27, 23 November 2023
of a subspace $A$ of a topological space $X$
A point $a\in A$ such that the intersection of some neighbourhood of $a$ with $A$ consists of the point $a$ alone.
A subset $A$ with no isolated points is dense-in-itself; a closed dense-in-itself subset is a perfect set.
References
| [1] | Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1978). Counterexamples in Topology (second edition). Berlin, New York: Springer-Verlag. ISBN 978-0-486-68735-3 MR507446 Zbl 0386.54001 |
How to Cite This Entry:
Isolated point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Isolated_point&oldid=33533
Isolated point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Isolated_point&oldid=33533
This article was adapted from an original article by A.A. Mal'tsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article