Georg Cantor at the Dawn of Point-Set Topology - Fourier Series and the Main Theorem

We present the argument of Cantor's main theorem in a way that is amenable to introducing the subject of point-set topology to junior or senior mathematics majors. The fact that point-set topology grew out of analysis and in particular trigonometric representation of a function immediately poses a difficulty in an historical introduction to point-set topology. A standard math curriculum usually places trigonometric representations of a function (Fourier series) in an upper level or second course in differential equations, a course that surprisingly few topology students take. Thus one would need to spend some time motivating interest in the question of trigonometric representation. Considering a trigonometric representation of a function as an analogous idea to a power series representation of a function from Calculus II can help the student see why it might be useful to represent a function in this way.

Because a power series is an infinite polynomial, a power series representation of a polynomial is neither interesting nor helpful. One application of Fourier series is that they allow us to represent polynomials in nontrivial ways and derive interesting results and formulas. For example, let \(f(x)=x\) for \({-\pi}<{x}<{\pi}.\) Then \(f(x)\) has Fourier series \[x=2\,\sum_{n=1}^{\infty}{\frac{{(-1)}^{n+1}}{n}}\sin{(nx)}.\] This representation for \(f(x)=x\) can be used (with \(x=\frac{\pi}{2}\)) to show Leibniz's famous formula for \(\pi;\) that is, \[{\frac{\pi}{4}}=1-{\frac{1}{3}}+{\frac{1}{5}}-{\frac{1}{7}}+\cdots\] Another fascinating formula that can be derived using Fourier series is the calculation of \[{\sum_{n=1}^{\infty}}\,{\frac{1}{n^2}}={\frac{{\pi}^2}{6}}\] by considering the expansion of \(f(x)=x^2\) on \({-\pi}\le{x}\le{\pi}.\) This function has Fourier series \[x^2={\frac{{\pi}^2}{3}}+4\,\,{{\sum_{n=1}^{\infty}}\,{\frac{{(-1)}^{n}}{n^2}}\cos{(nx)}}.\] and a substitution of \(x=\pi\) shows that \[\frac{{\pi}^2}{6}=\sum_{n=1}^{\infty}\,{\frac{1}{n^2}}.\] Furthermore, under reasonable hypotheses, one may differentiate a Fourier series term by term as one can a power series in calculus. The instructor might also mention some of the vast applications of Fourier series in physics [18].

Once the importance of such a representation has been established, the existence may be assumed and the question of the uniqueness of the representation is a natural question to consider. In order to show uniqueness, let \[{\frac{b_0}{2}}+\sum_{n=1}^{\infty}\,a_n\sin{(nx)} +b_n\cos{(nx)}\] and \[{\frac{b_0^{\prime}}{2}}+\sum_{i=1}^{\infty}\,{a^{\prime}_n}\sin{(nx)} +{b^{\prime}_n}\cos{(nx)}\] be two Fourier representations for a function \(f(x)\) on the interval \((0,2\pi).\) From here, we follow Cantor's argument, using his notation. If we set \(d_0 = b_0 - {b_0^{\prime}},\) \(c_n = a_n - {a^{\prime}_n},\) and \(d_n = b_n - {b^{\prime}_n},\) then \[0=\frac{d_0}{2}+\sum_{n=1}^{\infty}\,c_n\sin{(nx)} +d_n\cos{(nx)}.\] Cantor states his main theorem as follows and we see, finally, the role that a point set \(P\subseteq(0,2\pi)\) of the \(v\)th kind will play:

Theorem. If there is an equation of the form \[0 = C_0 + C_1 + C_2 +\cdots +C_n +\cdots ,\quad\quad\quad (2)\] where \[C_0 = \frac{d_0}{2};\,\,\,\,C_n = c_n\sin{(nx)} +d_n\cos{(nx)},\] for all values of \(x\) except those which correspond to points in the interval \(\left(0,2\pi\right)\) which give a point set \(P\) of the \(v\)th kind, where \(v\) signifies any large number, then \[d_0 =0,\quad c_n = d_n = 0.\]

Cantor begins his proof by defining \[F(x) = C_0\frac{x^2}{2}-C_1-\frac{C_2}{2^2}-\cdots -\frac{C_n}{n^2} -\cdots\] for all \(x\in(0,2\pi).\) Then \(F(x)\) is continuous, and if \(F(x)\) can be shown to be linear in the sense that \(F(x)= cx + c^\prime\) for real values \(c\) and \(c^{\prime},\) then Cantor may appeal to a result from his 1870 paper [4] to immediately conclude that \(c_n = d_n = 0,\) so that the Fourier series representation is unique. He is thus interested in conditions for which \(F(x)\) is linear.

Given the above set-up, Cantor showed in both his April 1870 paper [4] and his 1871 “Notiz” [5] that if Fourier series convergence is given up on a finite number of points in an interval \((p,q),\) then \(F(x)\) is linear on \((p,q).\) He restates this result in his 1872 paper, denoting by \(P\) the point set on which convergence 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.

Of course, if the set \(P\) is finite to begin with, result (A) guarantees that function \(F\) is linear on the interval \((0,2\pi),\) as desired. Cantor's challenge, then, was to extend result (A) from finite sets on which convergence (or representation) is given up to certain kinds of infinite sets on which convergence fails. More specifically, he set out to show that if an interval \((p,q)\) contains only finitely many points of the \(v\)th derived set \(P^{(v)}\) of \(P,\) then \(F\) is linear on that interval.