Accumulation point information

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 $x$ 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 some point of $S$ . Unlike for limit points, an adherent point $x$ of $S$ may have a neighbourhood not containing points other than $x$ 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.

With respect to the usual Euclidean topology, the sequence of rational numbers

x_{n}=(-1)^{n}{\frac {n}{n+1}}

has no limit (i.e. does not converge), but has two accumulation points (which are considered limit points here), viz. -1 and +1. Thus, thinking of sets, these points are limit points of the set

S=\{x_{n}\}.

^ Dugundji 1966, pp. 209–210.
^ Bourbaki 1989, pp. 68–83.
^ "Difference between boundary point & limit point". 2021-01-13.
^ "What is a limit point". 2021-01-13.
^ "Examples of Accumulation Points". 2021-01-13. Archived from the original on 2021-04-21. Retrieved 2021-01-14.

[FOOTNOTEDugundji1966209–210-1] Dugundji 1966, pp. 209–210.

[FOOTNOTEBourbaki198968–83-2] Bourbaki 1989, pp. 68–83.

[3] "Difference between boundary point & limit point". 2021-01-13.

[4] "What is a limit point". 2021-01-13.

[5] "Examples of Accumulation Points". 2021-01-13. Archived from the original on 2021-04-21. Retrieved 2021-01-14.

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