Neighborhood of a Point

(for a metric space), the set of all points whose distance from a given point is less than some positive number R. A neighborhood of this type is called spherical, and the number R is called the radius of the neighborhood. Also frequently considered are rectangular neighborhoods in a plane and their analogs in spaces of any number of dimensions. Sometimes the neighborhood of a point on a line is understood to mean any interval that includes this point, and the neighborhood of a point on a plane is understood to mean any open circle that includes this point but, perhaps, does not have the point as a center. These and other special types of neighborhoods are special cases of more general neighborhoods that are understood to mean all open sets that contain the given point.