Limit point

English

Noun

1. Template:Lb 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], Template:W, Analytic Function Theory, Volume 2, 2005, Template:W, page 19,

     The function

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

     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 $S^{\prime }$ be the set of limit points of a set $S$. Then the closure $\bar{S}$ of $S$ is $S\cup S^{\prime }$.

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     If $z_0$ is a limit point of $S$, then every $ϵ$-neighborhood of $z_0$ must contain infinitely many points of $S$.

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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 Template:M (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

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Translations

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See also

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Further reading

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