Isolated Point

a point belonging to some set M in whose neighborhood there is no other point of the set. The points of the set M that do not satisfy this condition are its limit points. The definition of an isolated point given above presupposes that in set Mthe concept of proximity among its elements (points) has been introduced. Because of this, the concept of isolated point is topological. In particular, if M is a set of points on a straight line, then point x of this set is an isolated point if there exists an interval that contains this point and does not contain other points of set M. Thus, if M consists of points having coordinates 1, 1/2, 1/3, …, l/n, …, then each point of this set is an isolated point, but for a set consisting of the same points and of a point having the coordinate 0, the latter will not be an isolated point. The isolated points of a curve or surface (here M is the set of all points of the given curve or surface) are also considered in geometry; for example, the point (0, 0) is an isolated point of the curve y² = x⁴ − 4 x². In the theory of functions of a complex variable we speak of the isolated singular points of an analytic function; the pole of a single-valued analytic function may serve as an example of this.