Difference between revisions of "Condensation point of a set"
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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. | 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. | ||
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====Comments==== | ====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 | + | 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 $x$ 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> | ||
Latest revision as of 09:56, 26 March 2023
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 $x$ 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=31620
Condensation point of a set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Condensation_point_of_a_set&oldid=31620
This article was adapted from an original article by B.A. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article