limit point

Noun

 * 1)  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.
 * 2) * 1962 [Ginn and Company],, Analytic Function Theory, Volume 2, 2005, , 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.
 * 1) * 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 $$S$$.
 * If $$z_0$$ is a limit point of $$S$$, then every $$\epsilon$$-neighborhood of $$z_0$$ must contain infinitely many points of $$S$$.

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 (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.

Translations

 * Dutch: verdichtingspunt
 * Finnish: rajapiste
 * French:
 * German:
 * Hebrew: נקודת גבול
 * Italian: punto di accumulazione
 * Polish: punkt skupienia