limit point

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limit point

Wikipedia

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 , Einar Hille, Analytic Function Theory, Volume 2, 2005, American Mathematical Society, page 19,

     The function

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

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

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

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

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

   • 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

See also

• limit

Further reading

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