# Accumulation point

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 ${\displaystyle x}$ with respect to the topology on ${\displaystyle X}$ also 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 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 a point of ${\displaystyle S}$ (that is, any point belonging to closure of the set). Unlike for limit points, an adherent point of ${\displaystyle S}$ may be ${\displaystyle x}$ 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).[3][4][5]