In this paper a continuum is a compact connected subset of the plane. When we consider two continua X and Y, one of the basic questions we ask is whether there exists a continuous map of X onto Y. More generally, when we consider two collections of continua, we ask when there exists a continuous map of a member of one collection onto a member of the other collection. We consider a collection of continua each of which is the union of the unit circle and a ray spiralling down upon the circle in a way to be defined later. The purpose of this paper is to determine which of these continua is the continuous image of a nonseparating continuum, i. e., a continuum that has a connected complement in the plane. This work is a special case of more general work done by David Bellamy. Bellamy proves the theorem for a larger class of "compactifications of (0,1] with remainder a circle" that are not assumed to be subsets of the plane, and he determines that the nonseparating continuum can be required to have the special property of being "chainable." Naturally his proof is more involved. We believe that our special case conveys the idea behind this result in a way accessible to undergraduate mathematics majors.
Reichel, Suzanne and Waldron, Kyen
"Sprials, Partial Sums and Continuous Images,"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 1
, Article 3.
Available at: http://scholar.rose-hulman.edu/rhumj/vol1/iss1/3