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 $(x_n)_$ 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 $\$ in $\R$ with standard topology. However, $0.5$ is a limit point (though not a boundary point) of interval $$, 1/math> 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.

Definition

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 \$ 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 $x_ = \left(x_n\right)_^$ 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 $x_$ if and only if $x$ is a limit of some subsequence of $x_.$
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 $x_ = \left(x_n\right)_^$ in $X$ is by definition just a map $x_ : \N \to X$ so that its image $\operatorname x_ := \left\$ 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 $\operatorname x_.$ For example, if the sequence is the constant sequence with value $x,$ we have $\operatorname x_ = \$ and $x$ is an isolated point of $\operatorname x_$ and not an accumulation point of $\operatorname x_.$

* 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 $\operatorname x_.$

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 $x_$ that will satisfy $A = \operatorname x_.$

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

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

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 $S.$

See also

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Citations

References

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{{Topology

Topology
General topology