"Limit point" redirects here. For uses where the word "point" is optional, see Limit (mathematics). For and, see Limit (disambiguation)#Mathematics.

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 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 \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).[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.

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.[6] 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

See also: Net (mathematics)#Cluster point of a net and Cluster point of a filter

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 limit point 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 not 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 definition 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.

Citations

  1. ^ Dugundji 1966, pp. 209–210.
  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.
  6. ^ Munkres 2000, pp. 97–102.