In mathematics, a limit point (or cluster point or accumulation point) of a set in a topological space is a point that can be "approximated" by points of in the sense that every neighbourhood of with respect to the topology on also contains a point of other than itself. A limit point of a set does not itself have to be an element of .

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 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 .

There is also a closely related concept for sequences. A cluster point (or accumulation point) of a sequence in a topological space is a point such that, for every neighbourhood of , there are infinitely many natural numbers such that . This concept generalizes to nets and filters.

DefinitionEdit

Let   be a subset of a topological space  . A point   in   is a limit point (or cluster point or accumulation point) of   if every neighbourhood of   contains at least one point of   different from   itself.

Note that it doesn't 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   is a   space (which all metric spaces are), then   is a limit point of   if and only if every neighbourhood of   contains infinitely many points of  . Indeed,   spaces are characterized by this property.

If   is a Fréchet–Urysohn space (which all metric spaces and first-countable spaces are), then   is a limit point of   if and only if there is a sequence of points in   whose limit is  . Indeed, Fréchet–Urysohn spaces are characterized by this property.

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

Types of limit pointsEdit

If every open set containing   contains infinitely many points of  , then   is a specific type of limit point called an  -accumulation point of  .

If every open set containing   contains uncountably many points of  , then   is a specific type of limit point called a condensation point of  .

If every open set   containing   satisfies  , then   is a specific type of limit point called a complete accumulation point of  .

For sequences and netsEdit

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

In a topological space  , a point   is said to be a cluster point (or accumulation point) of a sequence   if, for every neighbourhood   of  , there are infinitely many   such that  . It is equivalent to say that for every neighbourhood   of   and every  , there is some   such that  . If   is a metric space or a first-countable space (or, more generally, a Fréchet–Urysohn space), then   is cluster point of   if and only if   is a limit of some subsequence of  . The set of all cluster points of a sequence is sometimes called the limit set.

The concept of a net generalizes the idea of a sequence. A net is a function  , where   is a directed set and   is a topological space. A point   is said to be a cluster point (or accumulation point) of the net   if, for every neighbourhood   of   and every  , there is some   such that  , equivalently, if   has a subnet which converges to  . Cluster points in nets encompass the idea of both condensation points and ω-accumulation points. Clustering and limit points are also defined for the related topic of filters.

Selected factsEdit

  • We have the following characterization of limit points:   is a limit point of   if and only if it is in the closure of  .
    • Proof: We use the fact that a point is in the closure of a set if and only if every neighborhood of the point meets the set. Now,   is a limit point of  , if and only if every neighborhood of   contains a point of   other than  , if and only if every neighborhood of   contains a point of  , if and only if   is in the closure of  .
  • If we use   to denote the set of limit points of  , then we have the following characterization of the closure of  : The closure of   is equal to the union of   and  . This fact is sometimes taken as the definition of closure.
    • Proof: ("Left subset") Suppose   is in the closure of  . If   is in  , we are done. If   is not in  , then every neighbourhood of   contains a point of  , and this point cannot be  . In other words,   is a limit point of   and   is in  . ("Right subset") If   is in  , then every neighbourhood of   clearly meets  , so   is in the closure of  . If   is in  , then every neighbourhood of   contains a point of   (other than  ), so   is again in the closure of  . This completes the proof.
  • A corollary of this result gives us a characterisation of closed sets: A set   is closed if and only if it contains all of its limit points.
    • Proof:   is closed if and only if   is equal to its closure if and only if   if and only if   is contained in  .
    • Another proof: Let   be a closed set and   a limit point of  . If   is not in  , then the complement to   comprises an open neighbourhood of  . Since   is a limit point of  , any open neighbourhood of   should have a non-trivial intersection with  . However, a set can not have a non-trivial intersection with its complement. Conversely, assume   contains all its limit points. We shall show that the complement of   is an open set. Let   be a point in the complement of  . By assumption,   is not a limit point, and hence there exists an open neighbourhood U of   that does not intersect  , and so   lies entirely in the complement of  . Since this argument holds for arbitrary   in the complement of  , the complement of   can be expressed as a union of open neighbourhoods of the points in the complement of  . Hence the complement of   is open.
  • No isolated point is a limit point of any set.
    • Proof: If   is an isolated point, then   is a neighbourhood of   that contains no points other than  .
  • The closure   of a set   is a disjoint union of its limit points   and isolated points  :
 
  • A space   is discrete if and only if no subset of   has a limit point.
    • Proof: If   is discrete, then every point is isolated and cannot be a limit point of any set. Conversely, if   is not discrete, then there is a singleton   that is not open. Hence, every open neighbourhood of   contains a point  , and so   is a limit point of  .
  • If a space   has the trivial topology and   is a subset of   with more than one element, then all elements of   are limit points of  . If   is a singleton, then every point of   is still a limit point of  .
    • Proof: As long as  } is nonempty, its closure will be  . It's only empty when   is empty or   is the unique element of  .
  • By definition, every limit point is an adherent point.

See alsoEdit

ReferencesEdit

  • Hazewinkel, Michiel, ed. (2001) [1994], "Limit point of a set", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4

External linksEdit