# 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.