Topological Spaces and Jordan's Theorem

Abstract

What is topology? What is it based upon and what can I relate it to that I already know? These and many other questions were asked by those who attended the lecture given by J. Eldon Whitesitt, a guest lecturer from Montana State College, the second semester of last year. Curious, I resolved to find the answers to these questions about topology, that mysterious branch of mathematics, which differentiates between an orange and a doughnut by comparing the closed curves into which their respective geometric shapes can be broken. In the course of my investigation two key concepts continually appeared: the topological space and the Jordan curve. In this paper I should like to acquaint the reader with these ideas and try to relate them to some of the course material presented at Carroll. The first question that must fee answered is - "what is topology?" The etymology of the word infers that it is the study of place, but this is not strictly the case, at least not in a geometric sense. Topology is more strictly defined as the study of configurations of points which, when topological transformations are applied to them, remain unaffected by the transformation, that is, their essential properties are unaltered. But what is a topological transformation? It is a rule for mapping or identifying corresponding points in two different configurations such that each point of the original figure corresponds to one and only one point in the resulting figure, and such that mappings from the original figure to t e resulting one and back again are continuous.'' For example: f(x) = 2x maps every point on the real line into a corresponding point on the same or another real line, but maps no two points into the same point. This is what is known as a one to one mapping. The function is also continuous, since we shall define a continuous function to be one which maps open sets into open sets. The inverse of the function or as in our definition, the mapping back onto the original set of -points, written f-1, is also continuous by our definition. Thus, we can say that f(x) = 2x is a topological transformation of the real line onto the same or another real line.