limit point

English

Noun

limit point (plural limit points)

1. (topology) Given a subset S of a given topological space T, any point p whose every neighborhood contains some point, distinct from p, which belongs to S.
   • 1962 [Ginn and Company], Einar Hille, Analytic Function Theory, Volume 2, 2005, American Mathematical Society, page 19,

     The function

     $\sin \left(\cot {\frac {1}{z}}\right)$

     is an example of a function having infinitely many essential singularities with a limit point at $z=0$. It is an easy matter to give examples of "elementary" functions whose singularities have a countable number of limit points.

   • 1970 [Macmillan], John W. Dettman, Applied Complex Variables, 1984, Dover, unnumbered page,

     Let $S'$ be the set of limit points of a set $S$. Then the closure ${\overline {S}}$ of $S$ is $S\cup S'$.

     […]

     If $z_{0}$ is a limit point of $S$, then every $\epsilon$-neighborhood of $z_{0}$ must contain infinitely many points of $S$.

   • 2006, Ivan Francis Wilde, Lecture Notes on Complex Analysis, Imperial College Press, page 153:

     As the next example shows, the set of zeros may well have a limit point not belonging to the domain.

Usage notes

• The point p is called a limit point of S.
• Importantly, the limit point itself need not belong to S.
  • The union of S and the set of all limit points of S is called the closure (or topological closure) of S.
• If T is a T₁ space (a broad class that includes Hausdorff spaces and metric spaces), then the set of points in S in each neighborhood of a limit point p is at least countably infinite.

Synonyms

• (point whose every neighborhood contains a point in a given subset): accumulation point, cluster point

Translations

point whose every neighborhood contains a point in a given subset

• Dutch: verdichtingspunt
• Finnish: rajapiste
• French: point d’accumulation (fr) m
• German: Häufungspunkt (de) m
• Hebrew: נקודת גבול f (nekudát gvúl)
• Italian: punto di accumulazione m
• Polish: punkt skupienia m

See also

• limit

Further reading

• Closure (topology) on Wikipedia.Wikipedia
• Closed set on Wikipedia.Wikipedia