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Georg Cantor at the Dawn of Point-Set Topology - Point-Set Topology Has a History, Too!

Nicholas Scoville (Ursinus College)

First, a few words about why point-set topology is an important topic to understand from an historical point of view. Other authors have advanced many good reasons to study mathematics historically in general. Beyond these reasons, an historical approach to point-set topology should help a beginning student grasp and become interested in this area of mathematics, which is notoriously inaccessible to beginners. To the student, analysis is easily seen as a kind of "super-calculus" and abstract algebra can be motivated by discussing symmetries of objects. But when a course in point-set topology begins merely by defining a topology and giving several examples, it can be very difficult for students to grasp the general concept or see how topology connects with all the other mathematics they have learned. As we will see below, introducing point-set topology through its historical development motivates the student to consider the idea of "nearness without distance" as well as immediately places the subject within the larger mathematical world.

Point-set topology, which was originally called analysis situs or analysis of position, grew out of analysis. In discussing Cauchy's contribution to the foundations of analysis, Manheim writes that

The conceptual difficulties associated with the word limit derived from attempts to define it in terms of magnitude rather than aggregation. The unsatisfactory results of these endeavors led first to the formalism of Euler and later to that of Lagrange . . . [so] A new approach was required, an approach which recognized both the fundamental role of the limit concept and its basic arithmetical nature. [19, p. 26, emphasis original]

Here we see an attempt to isolate the limit concept from geometric and physical intuition. The now famous Weierstrass function [1, Section 5.4], which is both nowhere differentiable and everywhere continuous, verified that such an isolation was needed. Weierstrass lectured on this function in the 1860s and published a paper on it in 1872 [24]. Bolzano's paper proving the existence of a nowhere differentiable, everywhere continuous function was not published until 1930. Hyksová [15] nevertheless argues that Bolzano had constructed such a function as early as 1834. The famous Bolzano-Weierstrass Theorem today cannot be discussed without a knowledge of limit points. Although Weierstrass never defined limit point as a concept, it was not uncommon in his time to work with a concept without having an explicit definition for it [12]. Thus there are several places where one can see the arguable beginnings of point-set topology. Cantor's 1872 paper [6], the main focus of this article, is chosen as our object of study because it begins with a very clear and well-designed problem in analysis and solves this problem by introducing the derived set, a purely point-set notion. Thus it builds a “motivational bridge” between familiar concepts in analysis to a new concept in point-set topology, addressing the problem of point-set topology being disconnected from other branches of mathematics.

This paper has a two-fold purpose. First, it introduces the reader to Cantor's 1872 paper and in particular, his need for a theory of "nearness without distance" in order to solve an analysis problem. Second, the paper provides information to the teacher who would like to introduce point-set topology in an historical context, motivated by some of the questions that were popular at the time. Our two objectives are combined in Sections 7 and 8, where we present Cantor's main theorem in a way that should be accessible and provide motivation for the study of point-set topology. We begin with a brief discussion of the mathematical climate at the time of Cantor's 1872 paper. We use modern notation and parlance to convey Cantor's ideas, whenever doing so would make it easier for the modern reader of mathematics to understand and when there is no possibility of losing any of Cantor's original meaning or intent. (Editor's note: All passages from Cantor's 1872 paper that appear in this article were translated from German to English by the author.)

Nicholas Scoville (Ursinus College), "Georg Cantor at the Dawn of Point-Set Topology - Point-Set Topology Has a History, Too!," Convergence (May 2012), DOI:10.4169/loci003861