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Difference between revisions of "Accumulation point"

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''of a set'' $A$
 
''of a set'' $A$
  
 
A point $x$ in a topological space $X$ such that in any [[neighbourhood]] of $x$ there is a point of $A$ distinct from $x$. A set can have many accumulation points; on the other hand, it can have none. For example, any real number is an accumulation point of the set of all rational numbers in the ordinary topology. In a [[discrete space]], no set has an accumulation point. The set of all accumulation points of a set $A$ in a space $X$ is called the ''[[derived set]]'' (of $A$). In a $T_1$-space, every neighbourhood of an accumulation point of a set contains infinitely many points of the set.
 
A point $x$ in a topological space $X$ such that in any [[neighbourhood]] of $x$ there is a point of $A$ distinct from $x$. A set can have many accumulation points; on the other hand, it can have none. For example, any real number is an accumulation point of the set of all rational numbers in the ordinary topology. In a [[discrete space]], no set has an accumulation point. The set of all accumulation points of a set $A$ in a space $X$ is called the ''[[derived set]]'' (of $A$). In a $T_1$-space, every neighbourhood of an accumulation point of a set contains infinitely many points of the set.
  
The concept just defined should be distinguished from the concepts of a [[Proximate point|proximate point]] and a [[Complete accumulation point|complete accumulation point]]. In particular, any point of a set is a proximate point of the set, while it need not be an accumulation point (a counterexample: any point in a [[discrete space]]).
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The concept just defined should be distinguished from the concepts of a [[proximate point]] and a [[complete accumulation point]]. In particular, any point of a set is a proximate point of the set, while it need not be an accumulation point (a counterexample: any point in a [[discrete space]]).
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[[Category:General topology]]

Latest revision as of 16:48, 19 October 2014

of a set $A$

A point $x$ in a topological space $X$ such that in any neighbourhood of $x$ there is a point of $A$ distinct from $x$. A set can have many accumulation points; on the other hand, it can have none. For example, any real number is an accumulation point of the set of all rational numbers in the ordinary topology. In a discrete space, no set has an accumulation point. The set of all accumulation points of a set $A$ in a space $X$ is called the derived set (of $A$). In a $T_1$-space, every neighbourhood of an accumulation point of a set contains infinitely many points of the set.

The concept just defined should be distinguished from the concepts of a proximate point and a complete accumulation point. In particular, any point of a set is a proximate point of the set, while it need not be an accumulation point (a counterexample: any point in a discrete space).

How to Cite This Entry:
Accumulation point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Accumulation_point&oldid=30567
This article was adapted from an original article by A.V. Arkhangel'skii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article