Accumulation point

Template:Short description Template:Redirect In mathematics, a limit point, accumulation point, or cluster point of a set <math>S</math> in a topological space <math>X</math> is a point <math>x</math> that can be "approximated" by points of <math>S</math> in the sense that every neighbourhood of <math>x</math> contains a point of <math>S</math> other than <math>x</math> itself. A limit point of a set <math>S</math> does not itself have to be an element of <math>S.</math> There is also a closely related concept for sequences. A cluster point or accumulation point of a sequence <math>(x_n)_{n \in \N}</math> in a topological space <math>X</math> is a point <math>x</math> such that, for every neighbourhood <math>V</math> of <math>x,</math> there are infinitely many natural numbers <math>n</math> such that <math>x_n \in V.</math> This definition of a cluster or accumulation point of a sequence generalizes to nets and filters.

The similarly named notion of a Template:Em Template:Sfn (respectively, a limit point of a filter,Template:Sfn 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 Template:Em synonymous with "cluster/accumulation point of a sequence".

The limit points of a set should not be confused with adherent points (also called Template:Em) for which every neighbourhood of <math>x</math> contains some point of <math>S</math>. Unlike for limit points, an adherent point <math>x</math> of <math>S</math> may have a neighbourhood not containing points other than <math>x</math> 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, <math>0</math> is a boundary point (but not a limit point) of the set <math>\{0\}</math> in <math>\R</math> with standard topology. However, <math>0.5</math> is a limit point (though not a boundary point) of interval <math>[0, 1]</math> in <math>\R</math> with standard topology (for a less trivial example of a limit point, see the first caption).<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

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.

File:Rational sequence with 2 accumulation points.svg

With respect to the usual Euclidean topology, the sequence of rational numbers <math>x_n=(-1)^n \frac{n}{n+1}</math> has no Template:Em (i.e. does not converge), but has two accumulation points (which are considered Template:Em here), viz. -1 and +1. Thus, thinking of sets, these points are limit points of the set <math>S = \{x_n\}.</math>

DefinitionEdit

Accumulation points of a setEdit

File:Diagonal argument.svg

A sequence enumerating all positive rational numbers. Each positive real number is a cluster point.

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

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

The set of limit points of <math>S</math> is called the derived set of <math>S.</math>

Special types of accumulation point of a setEdit

If every neighbourhood of <math>x</math> contains infinitely many points of <math>S,</math> then <math>x</math> is a specific type of limit point called an Template:Visible anchor of <math>S.</math>

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

If every neighbourhood <math>U</math> of <math>x</math> is such that the cardinality of <math>U \cap S</math> equals the cardinality of <math>S,</math> then <math>x</math> is a specific type of limit point called a Template:Visible anchor of <math>S.</math>

Accumulation points of sequences and netsEdit

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In a topological space <math>X,</math> a point <math>x \in X</math> is said to be a Template:Visible anchor or Template:Visible anchor <math>x_{\bull} = \left(x_n\right)_{n=1}^{\infty}</math> if, for every neighbourhood <math>V</math> of <math>x,</math> there are infinitely many <math>n \in \N</math> such that <math>x_n \in V.</math> It is equivalent to say that for every neighbourhood <math>V</math> of <math>x</math> and every <math>n_0 \in \N,</math> there is some <math>n \geq n_0</math> such that <math>x_n \in V.</math> If <math>X</math> is a metric space or a first-countable space (or, more generally, a Fréchet–Urysohn space), then <math>x</math> is a cluster point of <math>x_{\bull}</math> if and only if <math>x</math> is a limit of some subsequence of <math>x_{\bull}.</math> 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 <math>x</math> to which the sequence converges (that is, every neighborhood of <math>x</math> contains all but finitely many elements of the sequence). That is why we do not use the term Template:Em 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 <math>f : (P,\leq) \to X,</math> where <math>(P,\leq)</math> is a directed set and <math>X</math> is a topological space. A point <math>x \in X</math> is said to be a [[Cluster point of a net|Template:Visible anchor]] or [[Accumulation point of a net|Template:Visible anchor]] <math>f</math> if, for every neighbourhood <math>V</math> of <math>x</math> and every <math>p_0 \in P,</math> there is some <math>p \geq p_0</math> such that <math>f(p) \in V,</math> equivalently, if <math>f</math> has a subnet which converges to <math>x.</math> 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 setEdit

Every sequence <math>x_{\bull} = \left(x_n\right)_{n=1}^{\infty}</math> in <math>X</math> is by definition just a map <math>x_{\bull} : \N \to X</math> so that its image <math>\operatorname{Im} x_{\bull} := \left\{ x_n : n \in \N \right\}</math> can be defined in the usual way.

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

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 <math>\omega</math>-accumulation point of the associated set <math>\operatorname{Im} x_{\bull}.</math>

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

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

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

PropertiesEdit

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 <math>\operatorname{cl}(S)</math> of a set <math>S</math> is a disjoint union of its limit points <math>L(S)</math> and isolated points <math>I(S)</math>; that is, <math display="block">\operatorname{cl} (S) = L(S) \cup I(S)\quad\text{and}\quad L(S) \cap I(S) = \emptyset.</math>

A point <math>x \in X</math> is a limit point of <math>S \subseteq X</math> if and only if it is in the closure of <math>S \setminus \{ x \}.</math> Template:Math proof

If we use <math>L(S)</math> to denote the set of limit points of <math>S,</math> then we have the following characterization of the closure of <math>S</math>: The closure of <math>S</math> is equal to the union of <math>S</math> and <math>L(S).</math> This fact is sometimes taken as the Template:Em of closure. Template:Math proof

A corollary of this result gives us a characterisation of closed sets: A set <math>S</math> is closed if and only if it contains all of its limit points. Template:Math proof

No isolated point is a limit point of any set. Template:Math proof

A space <math>X</math> is discrete if and only if no subset of <math>X</math> has a limit point. Template:Math proof

If a space <math>X</math> has the trivial topology and <math>S</math> is a subset of <math>X</math> with more than one element, then all elements of <math>X</math> are limit points of <math>S.</math> If <math>S</math> is a singleton, then every point of <math>X \setminus S</math> is a limit point of <math>S.</math> Template:Math proof

CitationsEdit

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ReferencesEdit

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