Georg Cantor at the Dawn of Point-Set Topology - Proof of the Main Theorem

At this point, it is worth outlining the important features to emphasize to the students.

• A trigonometric series representation for a function \(f(x)\) is important.
• Thus, the uniqueness of such a series representation is important.
• Cantor was able to show that if a certain function \(F(x)\) (based on \(f(x))\) is linear, then uniqueness follows.
• Cantor was able to show that if we give up convergence (or representation) for finitely many points, then \(F(x)\) is linear.

Having shown in previous papers that \(F\) is linear on \((0,2\pi)\) if convergence fails on a finite point set \(P,\) Cantor now sets out to show in his 1872 paper that if convergence fails on a possibly infinite point set \(P\) of the \(v\)th kind, then \(F\) still is linear on \((0,2\pi).\) This will establish his desired result, the uniqueness of the Fourier representation of the function \(f(x)\) on the interval \((0,2\pi).\)

As noted on page 7, Cantor begins by restating the theorem he proved in two previous papers. Again, \(P\) denotes the subset of \((0,2\pi)\) on which convergence of Fourier series for \(f(x)\) fails:

(A) If there is an interval \((p,q)\) in which only a finite number of points of the set \(P\) lie, then \(F(x)\) is linear in this interval.

Cantor now uses result (A) to establish result (A').

(A') If \((p^{\prime},q^{\prime})\) is any interval in which only a finite number of points of the set \(P^{\prime}\) lie, then \(F(x)\) is linear in this interval.

Remember that \(P^{\prime}\) is the derived set of \(P.\) Consider any interval \((p^{\prime},q^{\prime})\) which contains a finite number of points \({{x^{\prime}_0}}, {{x^{\prime}_1}},\dots,{{x^{\prime}_m}}\in P^{\prime},\) where \({{x^{\prime}_0}}<{{x^{\prime}_1}}<\cdots<{{x^{\prime}_m}}.\) Cantor will now apply result (A) to proper subintervals \((s,t)\) of the subinterval \(({{x^{\prime}_0}}, {{x^{\prime}_{1}}})\) with \({x^{\prime}_0} < s\) and \(t < {x^{\prime}_{1}}.\) We quote Cantor's argument, noting that he uses \(({{x^{\prime}_0}},\dots,{{x^{\prime}_1}})\) to denote the interval \(({{x^{\prime}_0}}, {{x^{\prime}_{1}}}).\)

Each of these subintervals generally contains infinitely many points of \(P\) so that result (A) does not directly apply; however each interval \((s,t)\) that falls within \(({{x^{\prime}_0}},\dots,{{x^{\prime}_1}})\) contains only a finite number of points from \(P\) (otherwise another point of the set \(P^{\prime}\) would fall between \({{x^{\prime}_0}}\) and \({{x^{\prime}_1}}\)), and the function is also linear on \((s,t)\) because of (A). The endpoints \(s\) and \(t\) can be made arbitrarily close to the points \({{x^{\prime}_0}}\) and \({{x^{\prime}_1}}\) so that the continuous function \(F(x)\) is also linear in \(({{x^{\prime}_0}},\dots,{{x^{\prime}_1}}).\)

Cantor illustrates this explanation with the diagram above. Of course his argument for the subinterval \(({{x^{\prime}_0}}, {{x^{\prime}_{1}}})\) applies to each of the subintervals \(({{x^{\prime}_i}}, {{x^{\prime}_{i+1}}})\) of \((p^{\prime},q^{\prime})\), and Cantor notes that it follows from the linearity of \(F\) on each of these subintervals that \(F\) is linear on \((p^{\prime},q^{\prime}),\) thereby establishing result (A').

In order to establish by mathematical induction the general result (A⁽ⁿ⁾), that, for every integer \(n\ge0,\) "If \((p^{(n)},q^{(n)})\) is any interval in which only a finite number of points of the set \(P^{(n)}\) lie, then \(F(x)\) is linear in this interval," Cantor notes that:

Once it is established that \(F(x)\) is a linear function on any interval \((p^{(k)},q^{(k)})\) with only a finite number of points from the \(k\)th derived point set \(P^{(k)}\) ..., it follows as in the (A) to (A') case that \(F(x)\) is a linear function on every interval \((p^{(k+1)},q^{(k+1)})\) which contains only a finite number of points of the \((k+1)\)th derived point set \(P^{(k+1)}.\)

Remember that Cantor's goal is to show that if convergence of Fourier series for a function \(f(x)\) fails on a possibly infinite point set \(P\subseteq(0,2\pi)\) of the \(v\)th kind, then the related function \(F\) still is linear on \((0,2\pi).\) This will establish his desired result, the uniqueness of the Fourier representation of the function \(f(x)\) on the interval \((0,2\pi).\) That \(P\) is a point-set of the \(v\)th kind means that \(P^{(v)}\) is finite (and \(P^{(v+1)}\) is empty), so that \((0,2\pi)\) contains only finitely many points of the \(v\)th derived set \(P^{(v)}.\) Result (A⁽ⁿ⁾) with n = \(v\) then guarantees that \(F\) is linear on \((0,2\pi),\) and hence that the Fourier representation of the function \(f(x)\) on \((0,2\pi)\) is unique.

A short presentation of the above theorem, along with the relevant definitions and emphasis on the four bullet points, should provide strong motivation for the study of concepts in point-set topology. My presentation of this material to students in a first course in point-set topology at Ursinus College during the spring semester of 2012 went as follows: We spent the first day of class going over the syllabus and discussing motivating examples. These examples included topology as "rubber sheet geometry" and "nearness without distance" (a concept I promised we would return to next class period), topology's applications to sensor networks (based on work of Professor Robert Ghrist of University of Pennsylvania), and the "invariance of domain" problem (the main goal of the course). As class ended, I gave the students a worksheet that we would discuss during the next class. This worksheet led the student through limit points, derived sets (including actual calculations), and Cantor's argument, as given above, motivated by the question of the uniqueness of a Fourier series representation. We spent the next class period answering questions and working through the exercises on the worksheet. We then talked through Cantor's argument while illustrating it with pictures on the board. I closed the discussion by emphasizing the fact that elements were part of the derived set not because they were necessarily "close" to other points in terms of distance, but because an open set around a point of a derived always has nonempty intersection with the original set, what one might term "nearness". This tied everything back to the previous class discussion about why one might wish to consider nearness without distance and what that even means! In the author's opinion, this approach gave the students a better understanding of where the course was going and why there is a need for such a deep level of abstraction.

We end this section with Roger Cooke’s summary [9]:

Seen in this context (rather than in the usual unmotivated classroom setting of point-set topology) the concept of a limit point and derived set are completely natural, almost inevitable, results of the attempt to decide the question of uniqueness of trigonometric series.