Georg Cantor at the Dawn of Point-Set Topology - Point-Set Topology

Author(s):
Nicholas Scoville (Ursinus College)

Before Cantor proves his main theorem, he gives several definitions which today we would recognize as belonging to the discipline of point-set topology. Recall Cantor's distinction between values and points above. He first defines a value set to be a finite or infinite set of values (number values). He then defines a point set to be a finite or infinite set of points. Modern mathematics tends to view the term "point-set" as synonymous with "open set." But Cantor's original understanding of point-set is any subset of the real line thought of as being in a one-to-one correspondence with a set of symbols on which you can "do" arithmetic. In fact, it is interesting to note, as G. H. Moore points out [20], that Cantor never used the idea of an open set. Today we define point-set topology in terms of open sets, yet the concept of open set as we know it took dozens of years to develop (again, see [20] for an excellent discussion of the history of open sets).

With a view towards generalizing his theorem, Cantor then defines a cluster point or limit point of a point set $$P$$ as

a point of the line situated in such a way that each neighborhood of it contains infinitely many points of $$P$$. (emphasis original)

This is the earliest known published definition of limit point. Cantor’s definition of a neighborhood of a point is "any interval that has the point as its interior" (emphasis original). (Interior point is not defined in this paper, but it would be defined in Cantor's 1879 paper [7].) Now that he has defined limit point, Cantor is able to partition the points of the real line into limit points of $$P$$ and non-limit points of $$P.$$ In this way, he defines the first derived set of $$P,$$ denoted $$P^{\prime},$$ to be the set of all limit points of $$P.$$ He may then define the second derived set of $$P,$$ denoted $$P^{\prime\prime},$$ as the first derived set of the first derived set $$P^{\prime}.$$ Continuing in this manner, Cantor defines $$P^{(v)},$$ the $$v$$th derived set of $$P,$$ noting that $$P^{(k)}$$ may be empty for some $$k.$$ This allows Cantor to define $$P$$ to be a point set of the $$v$$th kind whenever $$P^{(v)}$$ is finite (and hence $$P^{(v+1)}$$ is empty).

We construct a point set of the second kind. The reader can then intuit from this example how to construct a point set of the  $$v^{th}$$ kind for any $$v$$ (actually writing it down is messy). Let $$A_2=\left\{\frac{1}{n}+\frac{1}{2}: n>2, n\in \Bbb{N}\right\},$$ $$A_3=\left\{\frac{1}{n}+\frac{1}{3}:n>6, n\in \Bbb{N}\right\}, \ldots,$$ $$A_i=\left\{\frac{1}{n}+\frac{1}{i}: n>i(i-1), n\in \Bbb{N}\right\},\dots.$$ This last condition on $$n$$ ensures that $$A_i\subseteq\left[\frac{1}{i},\frac{1}{i-1}\right]$$ for $$i\geq2.$$ Define $$P=\bigcup\limits_{i=2}^{\infty}A_i.$$ Then $$P'=\left\{\frac{1}{n}: n\geq 2, n\in \Bbb{N}\right\}, P''=\{0\},\,\,{\rm and}\,\,P^{(3)}=\emptyset.$$

Nicholas Scoville (Ursinus College), "Georg Cantor at the Dawn of Point-Set Topology - Point-Set Topology," Loci (May 2012), DOI:10.4169/loci003861