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Difference between revisions of "Isolated point"

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''of a subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052770/i0527701.png" /> of a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052770/i0527702.png" />''
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''of a subspace $A$ of a topological space $X$''
  
A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052770/i0527703.png" /> such that the intersection of some [[Neighbourhood|neighbourhood]] of it with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052770/i0527704.png" /> consists of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052770/i0527705.png" /> alone.
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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.
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A subset $A$ with no isolated points is ''dense-in-itself''; a closed dense-in-itself subset is a ''[[perfect set]]''.
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====References====
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<table>
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<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>
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</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=13250
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