# Accumulation point

*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).

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Accumulation point.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Accumulation_point&oldid=33939