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

Nicholas Scoville (Ursinus College)

The primary focus of Cantor's paper is not point-set concepts. Rather, he was concerned with a certain theorem about Fourier series. Inspired by the work of Heine [13], Cantor was able to weaken conditions for which the Fourier series of a function is unique. He first did this in an 1870 paper [4] and, using the same technique, weakened the conditions further in an 1871 paper [5]. Both of these papers were precursors to his 1872 paper. For an in-depth discussion of the mathematics in these and Cantor's other papers around that same time, see Dauben [10]. Heine's 1870 work showed that if a function is almost everywhere continuous and its trigonometric series converges uniformly, then the Fourier series is unique. As Dauben points out,

Requiring almost-everywhere continuity and uniform convergence, Heine's theorem invited direct generalizations.

These generalizations would be taken up by Cantor. Important for our purposes is that Cantor developed a proof technique in his 1870 paper and modified it only slightly while weakening his hypotheses in the 1872 paper. More specifically, Cantor showed that, under certain hypotheses, the trigonometric representation of a function remains unique even when convergence or representation of the function is given up on certain infinite subsets of the open interval \((0,2\pi).\) (Cantor denoted the interval \((0,2\pi)\) by \((0,\dots,2\pi),\) but we will use modern notation.) What interests us in this article are the nature and construction of the particular kind of infinite set for which Cantor's 1872 theorem held. For here we see point-sets first defined, and in hindsight it is not surprising that point-sets soon came to be studied in their own right.