# Limit point

A point x in a topological space, all of whose neighborhoods contain some other point in a given subset that is different from x

This concept profitably generalizes the notion of a limit and is the underpinning of concepts such as closed set and topological closure. Indeed, a set is closed if and only if it contains all of its limit points, and the topological closure operation can be thought of as an operation that enriches a set by uniting it with its limit points.

It does not make a difference if we restrict the condition to open neighbourhoods only. It is often convenient to use the "open neighbourhood" form of the definition to show that a point is a limit point and to use the "general neighbourhood" form of the definition to derive facts from a known limit point.

Relation between accumulation point of a sequence and accumulation point of a set

Every limit of a non-constant sequence is an accumulation point of the sequence. And by definition, every limit point is an adherent point.