An accumulation point (also called a limit point or cluster point) of a set $S$ is a point $x$ such that every neighborhood around $x$ contains at least one point of $S$ different from $x$ itself. Importantly, $x$ does not have to belong to $S$. For example, $0$ is an accumulation point of the set ${1/n:n\in\mathbb{N}}$ because the elements of the set get arbitrarily close to $0$. More generally, accumulation points describe locations that can be approached by points of a set, making the concept fundamental to limits and topology.
Accumulation points are closely connected to closure and closed sets. The closure of a set consists of the original set together with all of its accumulation points, and a set is closed exactly when it contains all of its accumulation points. For sequences, an accumulation point is a point such that every neighborhood contains infinitely many terms of the sequence; in metric spaces, this is equivalent to being the limit of some subsequence. A sequence can therefore have several accumulation points even when it does not converge—for example, the sequence $(-1)^n n/(n+1)$ has accumulation points $-1$ and $1$.
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Accumulation points are closely connected to closure and closed sets. The closure of a set consists of the original set together with all of its accumulation points, and a set is closed exactly when it contains all of its accumulation points. For sequences, an accumulation point is a point such that every neighborhood contains infinitely many terms of the sequence; in metric spaces, this is equivalent to being the limit of some subsequence. A sequence can therefore have several accumulation points even when it does not converge—for example, the sequence $(-1)^n n/(n+1)$ has accumulation points $-1$ and $1$.
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