limitpoint (plurallimit points)

1. (topology) Given a subsetS of a giventopological spaceT, any pointp whose every neighborhood contains some point, distinct fromp, which belongs toS.
   • 1962 [Ginn and Company],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 alimit point at${\displaystyle z=0}$ . It is an easy matter to give examples of "elementary" functions whose singularities have a countable number oflimit points.

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

     Let${\displaystyle S^{\prime }}$  be the set oflimit points of a set${\displaystyle S}$ . Then the closure${\displaystyle \overline{S}}$  of${\displaystyle S}$  is${\displaystyle S\cup S^{\prime }}$ .

     […]

     If${\displaystyle z_0}$  is alimit point of${\displaystyle S}$ , then every${\displaystyle ϵ}$ -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,page153:

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

• The pointp is called alimit point of S.
• Importantly, the limit point itself need not belong toS.
  • Theunion ofS and the set of all limit points ofS is called theclosure (ortopological closure) ofS.
• IfT is aT₁ space (a broad class that includesHausdorff spaces andmetric spaces), then the set of points inS in each neighborhood of a limit pointp is at leastcountably infinite.

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

• limit

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