Namespaces
Variants
Actions

Difference between revisions of "Isolated point"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (better)
(3 intermediate revisions by 2 users not shown)
Line 1: Line 1:
''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" />''
+
{{TEX|done}}
 +
''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.
+
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-in-itself''; a closed dense-in-itself subset is a ''[[perfect set]]''.
 +
 
 +
====References====
 +
<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. MR 507446.  Zbl 0386.54001.</TD></TR>
 +
</table>

Revision as of 16:43, 11 October 2014

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. MR 507446. 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