Accumulation point

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 $$(x_n)_{n \in \N}$$ 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 (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 synonymous with "cluster/accumulation point of a sequence".

The limit points of a set should not be confused with adherent points (also called ) 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 $$\R$$ with standard topology. However, $$0.5$$ is a limit point (though not a boundary point) of interval $$[0, 1]$$ in $$\R$$ 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.



Accumulation points of a set


Let $$S$$ be a subset of a topological space $$X.$$ A point $$x$$ in $$X$$ is a limit point or cluster point or  $$S$$ if every neighbourhood of $$x$$ contains at least one point of $$S$$ different from $$x$$ 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 $$X$$ is a $T_1$ space (such as a metric space), then $$x \in X$$ is a limit point of $$S$$ if and only if every neighbourhood of $$x$$ contains infinitely many points of $$S.$$ In fact, $$T_1$$ spaces are characterized by this property.

If $$X$$ is a Fréchet–Urysohn space (which all metric spaces and first-countable spaces are), then $$x \in X$$ is a limit point of $$S$$ if and only if there is a sequence of points in $$S \setminus \{x\}$$ whose limit is $$x.$$ In fact, Fréchet–Urysohn spaces are characterized by this property.

The set of limit points of $$S$$ is called the derived set of $$S.$$

Special types of accumulation point of a set
If every neighbourhood of $$x$$ contains infinitely many points of $$S,$$ then $$x$$ is a specific type of limit point called an  of $$S.$$

If every neighbourhood of $$x$$ contains uncountably many points of $$S,$$ then $$x$$ is a specific type of limit point called a condensation point of $$S.$$

If every neighbourhood $$U$$ of $$x$$ is such that the cardinality of $$U \cap S$$ equals the cardinality of $$S,$$ then $$x$$ is a specific type of limit point called a  of $$S.$$

Accumulation points of sequences and nets
In a topological space $$X,$$ a point $$x \in X$$ is said to be a ' or ' $$x_{\bull} = \left(x_n\right)_{n=1}^{\infty}$$ if, for every neighbourhood $$V$$ of $$x,$$ there are infinitely many $$n \in \N$$ such that $$x_n \in V.$$ It is equivalent to say that for every neighbourhood $$V$$ of $$x$$ and every $$n_0 \in \N,$$ there is some $$n \geq n_0$$ such that $$x_n \in V.$$ If $$X$$ is a metric space or a first-countable space (or, more generally, a Fréchet–Urysohn space), then $$x$$ is a cluster point of $$x_{\bull}$$ if and only if $$x$$ is a limit of some subsequence of $$x_{\bull}.$$ The set of all cluster points of a sequence is sometimes called the limit set.

Note that there is already the notion of limit of a sequence to mean a point $$x$$ to which the sequence converges (that is, every neighborhood of $$x$$ contains all but finitely many elements of the sequence). That is why we do not use the term of a sequence as a synonym for accumulation point of the sequence.

The concept of a net generalizes the idea of a sequence. A net is a function $$f : (P,\leq) \to X,$$ where $$(P,\leq)$$ is a directed set and $$X$$ is a topological space. A point $$x \in X$$ is said to be a ' or ' $$f$$ if, for every neighbourhood $$V$$ of $$x$$ and every $$p_0 \in P,$$ there is some $$p \geq p_0$$ such that $$f(p) \in V,$$ equivalently, if $$f$$ has a subnet which converges to $$x.$$ Cluster points in nets encompass the idea of both condensation points and ω-accumulation points. Clustering and limit points are also defined for filters.

Relation between accumulation point of a sequence and accumulation point of a set
Every sequence $$x_{\bull} = \left(x_n\right)_{n=1}^{\infty}$$ in $$X$$ is by definition just a map $$x_{\bull} : \N \to X$$ so that its image $$\operatorname{Im} x_{\bull} := \left\{ x_n : n \in \N \right\}$$ can be defined in the usual way.


 * If there exists an element $$x \in X$$ that occurs infinitely many times in the sequence, $$x$$ is an accumulation point of the sequence. But $$x$$ need not be an accumulation point of the corresponding set $$\operatorname{Im} x_{\bull}.$$ For example, if the sequence is the constant sequence with value $$x,$$ we have $$\operatorname{Im} x_{\bull} = \{ x \}$$ and $$x$$ is an isolated point of $$\operatorname{Im} x_{\bull}$$ and not an accumulation point of $$\operatorname{Im} x_{\bull}.$$


 * If no element occurs infinitely many times in the sequence, for example if all the elements are distinct, any accumulation point of the sequence is an $$\omega$$-accumulation point of the associated set $$\operatorname{Im} x_{\bull}.$$

Conversely, given a countable infinite set $$A \subseteq X$$ in $$X,$$ we can enumerate all the elements of $$A$$ in many ways, even with repeats, and thus associate with it many sequences $$x_{\bull}$$ that will satisfy $$A = \operatorname{Im} x_{\bull}.$$


 * Any $$\omega$$-accumulation point of $$A$$ is an accumulation point of any of the corresponding sequences (because any neighborhood of the point will contain infinitely many elements of $$A$$ and hence also infinitely many terms in any associated sequence).


 * A point $$x \in X$$ that is an $$\omega$$-accumulation point of $$A$$ cannot be an accumulation point of any of the associated sequences without infinite repeats (because $$x$$ has a neighborhood that contains only finitely many (possibly even none) points of $$A$$ and that neighborhood can only contain finitely many terms of such sequences).

Properties
Every limit of a non-constant sequence is an accumulation point of the sequence. And by definition, every limit point is an adherent point.

The closure $$\operatorname{cl}(S)$$ of a set $$S$$ is a disjoint union of its limit points $$L(S)$$ and isolated points $$I(S)$$; that is, $$\operatorname{cl} (S) = L(S) \cup I(S)\quad\text{and}\quad L(S) \cap I(S) = \emptyset.$$

A point $$x \in X$$ is a limit point of $$S \subseteq X$$ if and only if it is in the closure of $$S \setminus \{ x \}.$$ $$

If we use $$L(S)$$ to denote the set of limit points of $$S,$$ then we have the following characterization of the closure of $$S$$: The closure of $$S$$ is equal to the union of $$S$$ and $$L(S).$$ This fact is sometimes taken as the of closure. $$

A corollary of this result gives us a characterisation of closed sets: A set $$S$$ is closed if and only if it contains all of its limit points. $$

No isolated point is a limit point of any set. $$

A space $$X$$ is discrete if and only if no subset of $$X$$ has a limit point. $$

If a space $$X$$ has the trivial topology and $$S$$ is a subset of $$X$$ with more than one element, then all elements of $$X$$ are limit points of $$S.$$ If $$S$$ is a singleton, then every point of $$X \setminus S$$ is a limit point of $$S.$$ $$