Filter on a set in the context of "Limit 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 ${\displaystyle x}$ 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.