Limit Point - Types of Limit Points

Types of Limit Points

If every open set containing x contains infinitely many points of S then x is a specific type of limit point called a ω-accumulation point of S.

If every open set containing x contains uncountably many points of S then x is a specific type of limit point called a condensation point of S.

If every open set U containing x satisfies |U ∩ S| = |S| then x is a specific type of limit point called a complete accumulation point of S.

A point x ∈ X is a cluster point or accumulation point of a sequence (x_n)_{n ∈ N} if, for every neighbourhood V of x, there are infinitely many natural numbers n such that x_n ∈ V. If the space is sequential, this is equivalent to the assertion that x is a limit of some subsequence of the sequence (x_n)_{n ∈ N}.

The concept of a net generalizes the idea of a sequence. Cluster points in nets encompass the idea of both condensation points and ω-accumulation points. Clustering and limit points are also defined for the related topic of filters.

The set of all cluster points of a sequence is sometimes called a limit set.