Point-Set Topology

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 $$P'=\left\{\frac{1}{n}: n\geq 2, n\in \Bbb{N}\right\}, P''=\{0\},\,\,{\rm and}\,\,P^{(3)}=\emptyset.$$