# Accumulation point

## Cluster point in a topological space / From Wikipedia, the free encyclopedia

In mathematics, a **limit point**, **accumulation point**, or **cluster 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 $S.$
There is also a closely related concept for sequences. A **cluster point** or **accumulation point** of a sequence $(x_{n})_{n\in \mathbb {N} }$ 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 $x_{n}\in V.$ This definition of a cluster or accumulation point of a sequence generalizes to nets and filters.

The similarly named notion of a *limit point of a sequence*[1] (respectively, a limit point of a filter,[2] a limit point of a net) by definition refers to a point that the sequence converges to (respectively, the filter converges to, the net converges to). Importantly, although "limit point of a set" is synonymous with "cluster/accumulation point of a set", this is not true for sequences (nor nets or filters). That is, the term "limit point of a sequence" is *not* synonymous with "cluster/accumulation point of a sequence".

The limit points of a set should not be confused with adherent points (also called *points of closure*) for which every neighbourhood of $x$ contains a point of $S$ (that is, any point belonging to closure of the set). Unlike for limit points, an adherent point of $S$ may be $x$ itself. A limit point can be characterized as an adherent point that is not an isolated point.

Limit points of a set should also not be confused with boundary points. For example, $0$ is a boundary point (but not a limit point) of the set $\{0\}$ in $\mathbb {R}$ with standard topology. However, $0.5$ is a limit point (though not a boundary point) of interval $[0,1]$ in $\mathbb {R}$ with standard topology (for a less trivial example of a limit point, see the first caption).[3][4][5]

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.