Limit Points and Connected Sets in the Plane

Editor’s note: This article was published in August 2007.

Intuitively a set S in the plane is connected if for every pair of distinct points p₁ = (x₁,y₁) and p₂ = (x₂,y₂) and in S there is a curve consisting of points in S with endpoints p₁ and p₂.  This definition was sufficient until the end of the nineteenth century, when an implosion of imaginative examples of curves in the plane demanded a more rigorous definition.  One purpose of this paper is to trace this aspect of the early history of connected sets based on the formal definition that replaced the intuitive one.  To illustrate the need for such a rigorous definition, we introduce a curve called Mullikin’s nautilus that was first constructed in 1919.  Since the formal definition relies on limit points, the nautilus not only illustrates its necessity but serves as an instructive vehicle for introducing limits in the plane to students who have studied limits on the real line.  Therefore the second purpose of this paper is to present material directly applicable to the classroom.