cluster point

A cluster point of a set is a point that the set crowds around: no matter how tightly you zoom in, you always find other members of the set huddled nearby. It is a place you can sneak up on using points of the set, getting arbitrarily close without ever needing to land on it.

Precisely, p is a cluster point of a set S if every neighborhood of p contains at least one point of S different from p itself — equivalently, every interval (p - r, p + r) meets S in a point other than p. An equivalent sequential form: there is a sequence of points x_n in S, all different from p, with x_n -> p. The point p may or may not belong to S.

Cluster points are exactly the places where it makes sense to take a limit of a function: lim_{x->a} f(x) is only defined when a is a cluster point of the domain, so that there really are inputs arbitrarily close to a to approach along. They are the opposite of isolated points — a point of a set is either isolated in it or a cluster point of it, never both.