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Difference between revisions of "Condensation point of a set"

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''in a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024440/c0244401.png" />''
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''in a Euclidean space $E^n$''
  
A point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024440/c0244402.png" /> such that every neighbourhood of it contains uncountable many points of the set. The set of condensation points of a set is always closed; if it is non-empty, it is perfect and has the cardinality of the continuum. The concept of a condensation point can be generalized to arbitrary topological spaces.
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A point of $E^n$ such that every neighbourhood of it contains uncountable many points of the set. The set of condensation points of a set is always closed; if it is non-empty, it is perfect and has the cardinality of the continuum. The concept of a condensation point can be generalized to arbitrary topological spaces.
  
  
  
 
====Comments====
 
====Comments====
The generalization to arbitrary spaces is direct: A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024440/c0244403.png" /> a condensation point (of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024440/c0244404.png" />) in a topological space if (the intersection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024440/c0244405.png" /> with) every neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024440/c0244406.png" /> is an uncountable set. (See also [[#References|[a1]]].)
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The generalization to arbitrary spaces is direct: A point $x$ a condensation point (of a set $M$) in a topological space if (the intersection of $M$ with) every neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024440/c0244406.png" /> is an uncountable set. (See also [[#References|[a1]]].)
  
 
====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)  (Translated from Russian)</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)  (Translated from Russian)</TD></TR></table>

Revision as of 12:13, 12 April 2014

in a Euclidean space $E^n$

A point of $E^n$ such that every neighbourhood of it contains uncountable many points of the set. The set of condensation points of a set is always closed; if it is non-empty, it is perfect and has the cardinality of the continuum. The concept of a condensation point can be generalized to arbitrary topological spaces.


Comments

The generalization to arbitrary spaces is direct: A point $x$ a condensation point (of a set $M$) in a topological space if (the intersection of $M$ with) every neighbourhood of is an uncountable set. (See also [a1].)

References

[a1] A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian)
How to Cite This Entry:
Condensation point of a set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Condensation_point_of_a_set&oldid=12877
This article was adapted from an original article by B.A. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article