In mathematics, a limit point, accumulation point, or cluster point of a set ${\displaystyle S}$ in a topological space ${\displaystyle X}$ is a point ${\displaystyle x}$ that can be "approximated" by points of ${\displaystyle S}$ in the sense that every neighbourhood of ${\displaystyle x}$ contains a point of ${\displaystyle S}$ other than ${\displaystyle x}$ itself. A limit point of a set ${\displaystyle S}$ does not itself have to be an element of ${\displaystyle S.}$ There is also a closely related concept for sequences. A cluster point or accumulation point of a sequence ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ in a topological space ${\displaystyle X}$ is a point ${\displaystyle x}$ such that, for every neighbourhood ${\displaystyle V}$ of ${\displaystyle x,}$ there are infinitely many natural numbers ${\displaystyle n}$ such that ${\displaystyle x_{n}\in V.}$ 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 (respectively, a limit point of a filter, 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 ${\displaystyle x}$ contains some point of ${\displaystyle S}$. Unlike for limit points, an adherent point ${\displaystyle x}$ of ${\displaystyle S}$ may have a neighbourhood not containing points other than ${\displaystyle x}$ 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, ${\displaystyle 0}$ is a boundary point (but not a limit point) of the set ${\displaystyle \{0\}}$ in ${\displaystyle \mathbb {R} }$ with standard topology. However, ${\displaystyle 0.5}$ is a limit point (though not a boundary point) of interval ${\displaystyle [0,1]}$ in ${\displaystyle \mathbb {R} }$ with standard topology (for a less trivial example of a limit point, see the first caption).

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.

Let ${\displaystyle S}$ be a subset of a topological space ${\displaystyle X.}$ A point ${\displaystyle x}$ in ${\displaystyle X}$ is a limit point or cluster point or accumulation point of the set ${\displaystyle S}$ if every neighbourhood of ${\displaystyle x}$ contains at least one point of ${\displaystyle S}$ different from ${\displaystyle x}$ itself.

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.

If ${\displaystyle X}$ is a ${\displaystyle T_{1}}$ space (such as a metric space), then ${\displaystyle x\in X}$ is a limit point of ${\displaystyle S}$ if and only if every neighbourhood of ${\displaystyle x}$ contains infinitely many points of ${\displaystyle S.}$ In fact, ${\displaystyle T_{1}}$ spaces are characterized by this property.