Georg Cantor at the Dawn of Point-Set Topology

Overview

A first course in point-set topology can be challenging for the student because of the abstract level of the material. In an attempt to mitigate this problem, we use the history of point-set topology to obtain natural motivation for the study of some key concepts. In this article, we study an 1872 paper by Georg Cantor. We will look at the problem Cantor was attempting to solve and see how the now familiar concepts of a point-set and derived set are natural answers to his question. We emphasize ways to utilize Cantor's methods in order to introduce point-set topology to students.

Introduction

In his introduction to his book Introduction to Phenomenology [23], Msgr. Robert Sokolowski writes

Mathematicians . . . tend to absorb the writings of their predecessors directly into their own work. They do not comment on the writings of earlier mathematicians, even if they have been very much influenced by them. They simply make use of the material that they find in the authors they read. When advances are made in mathematics, later thinkers condense the findings and move on. Few mathematicians study works from past centuries; compared with contemporary mathematics, such older writings seem to them almost like the work of children. 

As a philosopher, Msgr. Sokolowski is accustomed to the traditional methods of teaching philosophy to undergraduates – start with Plato, Aristotle and the other ancients, continue with developments through the Scholastic and Enlightenment eras, and then show how modern philosophy builds upon all that has gone before. He must be puzzled, then, by the lack of attention to the historical development of ideas that generally attends to the teaching of mathematics. He perceives that something important is missing, and he is correct.

In recent years, interest has grown considerably in developing an historical approach to the teaching of mathematics. Victor Katz has edited an anthology of articles giving different perspectives on the development of mathematics in general from an historical point of view [16]. Some authors, such as Klyve, Stemkoski, and Tou, focus on one particular historical figure – in their case, Euler – important to the development of mathematics [17]. There is also interest in the historical development of certain areas of mathematics commonly included in the undergraduate curriculum. Brian Hopkins has written a textbook introducing discrete mathematics from an historical point of view [14]; David Bressoud has written two textbooks that present analysis from an historical perspective ([2], [3]); and Adam Parker has compiled an original sources bibliography for ordinary differential equations instructors that contains many of the original papers in ODEs.

This is the first paper in a planned series that will outline ways to introduce point-set topology concepts motivated by their place in history. To borrow a phrase from David Bressoud, it is an "attempt to let history inform pedagogy" [2, p. vii]. A growing collection of the historic papers that are important to the development of point-set topology may be found on the author's web site.

This paper focuses on the seminal work of Georg Cantor (1845-1918), a German mathematician well-known for his contributions to the foundations of set theory, but whose contributions to point-set topology are not very well known. Cantor’s works are collected in [8]. For complete biographical information, see Dauben’s definitive work [11].