Limit point

In mathematics, a limit point (or cluster point or accumulation point) of a set S in a topological space X is a point x that can be 'approximated' by points of S in the sense that every neighbourhood of x with respect to the topology on X also contains a point of S other than x itself. A limit point of a set S does not itself have to be an element of S. In mathematics, a limit point (or cluster point or accumulation point) of a set S in a topological space X is a point x that can be 'approximated' by points of S in the sense that every neighbourhood of x with respect to the topology on X also contains a point of S other than x itself. A limit point of a set S does not itself have to be an element of S. 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. There is also a closely related concept for sequences. A cluster point (or accumulation point) of a sequence (xn)n ∈ N in a topological space X is a point x such that, for every neighbourhood V of x, there are infinitely many natural numbers n such that xn ∈ V. This concept generalizes to nets and filters. Let S be a subset of a topological space X. A point x in X is a limit point (or cluster point or accumulation point) of S if every neighbourhood of x contains at least one point of S different from x itself.