# Accumulation point

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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=33939
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