**Overview**

A first course in point-set topology can be challenging for the student because of the abstract level of the material. In an attempt to mitigate this problem, we use the history of point-set topology to obtain natural motivation for the study of some key concepts. In this article, we study an 1872 paper by Georg Cantor. We will look at the problem Cantor was attempting to solve and see how the now familiar concepts of a point-set and derived set are natural answers to his question. We emphasize ways to utilize Cantor's methods in order to introduce point-set topology to students.

**Introduction**

In his introduction to his book *Introduction to Phenomenology* [23], Msgr. Robert Sokolowski writes

Mathematicians . . . tend to absorb the writings of their predecessors directly into their own work. They do not comment on the writings of earlier mathematicians, even if they have been very much influenced by them. They simply make use of the material that they find in the authors they read. When advances are made in mathematics, later thinkers condense the findings and move on. Few mathematicians study works from past centuries; compared with contemporary mathematics, such older writings seem to them almost like the work of children.

As a philosopher, Msgr. Sokolowski is accustomed to the traditional methods of teaching philosophy to undergraduates – start with Plato, Aristotle and the other ancients, continue with developments through the Scholastic and Enlightenment eras, and then show how modern philosophy builds upon all that has gone before. He must be puzzled, then, by the lack of attention to the historical development of ideas that generally attends to the teaching of mathematics. He perceives that something important is missing, and he is correct.

In recent years, interest has grown considerably in developing an historical approach to the teaching of mathematics. Victor Katz has edited an anthology of articles giving different perspectives on the development of mathematics in general from an historical point of view [16]. Some authors, such as Klyve, Stemkoski, and Tou, focus on one particular historical figure – in their case, Euler – important to the development of mathematics [17]. There is also interest in the historical development of certain areas of mathematics commonly included in the undergraduate curriculum. Brian Hopkins has written a textbook introducing discrete mathematics from an historical point of view [14]; David Bressoud has written two textbooks that present analysis from an historical perspective ([2], [3]); and Adam Parker has compiled an original sources bibliography for ordinary differential equations instructors that contains many of the original papers in ODEs.

This is the first paper in a planned series that will outline ways to introduce point-set topology concepts motivated by their place in history. To borrow a phrase from David Bressoud, it is an "attempt to let history inform pedagogy" [2, p. vii]. A growing collection of the historic papers that are important to the development of point-set topology may be found on the author's web site.

This paper focuses on the seminal work of Georg Cantor (1845-1918), a German mathematician well-known for his contributions to the foundations of set theory, but whose contributions to point-set topology are not very well known. Cantor’s works are collected in [8]. For complete biographical information, see Dauben’s definitive work [11].

**Overview**

A first course in point-set topology can be challenging for the student because of the abstract level of the material. In an attempt to mitigate this problem, we use the history of point-set topology to obtain natural motivation for the study of some key concepts. In this article, we study an 1872 paper by Georg Cantor. We will look at the problem Cantor was attempting to solve and see how the now familiar concepts of a point-set and derived set are natural answers to his question. We emphasize ways to utilize Cantor's methods in order to introduce point-set topology to students.

**Introduction**

In his introduction to his book *Introduction to Phenomenology* [23], Msgr. Robert Sokolowski writes

Mathematicians . . . tend to absorb the writings of their predecessors directly into their own work. They do not comment on the writings of earlier mathematicians, even if they have been very much influenced by them. They simply make use of the material that they find in the authors they read. When advances are made in mathematics, later thinkers condense the findings and move on. Few mathematicians study works from past centuries; compared with contemporary mathematics, such older writings seem to them almost like the work of children.

As a philosopher, Msgr. Sokolowski is accustomed to the traditional methods of teaching philosophy to undergraduates – start with Plato, Aristotle and the other ancients, continue with developments through the Scholastic and Enlightenment eras, and then show how modern philosophy builds upon all that has gone before. He must be puzzled, then, by the lack of attention to the historical development of ideas that generally attends to the teaching of mathematics. He perceives that something important is missing, and he is correct.

In recent years, interest has grown considerably in developing an historical approach to the teaching of mathematics. Victor Katz has edited an anthology of articles giving different perspectives on the development of mathematics in general from an historical point of view [16]. Some authors, such as Klyve, Stemkoski, and Tou, focus on one particular historical figure – in their case, Euler – important to the development of mathematics [17]. There is also interest in the historical development of certain areas of mathematics commonly included in the undergraduate curriculum. Brian Hopkins has written a textbook introducing discrete mathematics from an historical point of view [14]; David Bressoud has written two textbooks that present analysis from an historical perspective ([2], [3]); and Adam Parker has compiled an original sources bibliography for ordinary differential equations instructors that contains many of the original papers in ODEs.

This is the first paper in a planned series that will outline ways to introduce point-set topology concepts motivated by their place in history. To borrow a phrase from David Bressoud, it is an "attempt to let history inform pedagogy" [2, p. vii]. A growing collection of the historic papers that are important to the development of point-set topology may be found on the author's web site.

This paper focuses on the seminal work of Georg Cantor (1845-1918), a German mathematician well-known for his contributions to the foundations of set theory, but whose contributions to point-set topology are not very well known. Cantor’s works are collected in [8]. For complete biographical information, see Dauben’s definitive work [11].

First, a few words about why point-set topology is an important topic to understand from an historical point of view. Other authors have advanced many good reasons to study mathematics historically in general. Beyond these reasons, an historical approach to point-set topology should help a beginning student grasp and become interested in this area of mathematics, which is notoriously inaccessible to beginners. To the student, analysis is easily seen as a kind of "super-calculus" and abstract algebra can be motivated by discussing symmetries of objects. But when a course in point-set topology begins merely by defining a topology and giving several examples, it can be very difficult for students to grasp the general concept or see how topology connects with all the other mathematics they have learned. As we will see below, introducing point-set topology through its historical development motivates the student to consider the idea of "nearness without distance" as well as immediately places the subject within the larger mathematical world.

Point-set topology, which was originally called *analysis situs* or analysis of position, grew out of analysis. In discussing Cauchy's contribution to the foundations of analysis, Manheim writes that

The conceptual difficulties associated with the word

limitderived from attempts to define it in terms of magnitude rather than aggregation. The unsatisfactory results of these endeavors led first to the formalism of Euler and later to that of Lagrange . . . [so] A new approach was required, an approach which recognized both the fundamental role of the limit concept and its basic arithmetical nature. [19, p. 26, emphasis original]

Here we see an attempt to isolate the limit concept from geometric and physical intuition. The now famous Weierstrass function [1, Section 5.4], which is both nowhere differentiable and everywhere continuous, verified that such an isolation was needed. Weierstrass lectured on this function in the 1860s and published a paper on it in 1872 [24]. Bolzano's paper proving the existence of a nowhere differentiable, everywhere continuous function was not published until 1930. Hyksová [15] nevertheless argues that Bolzano had constructed such a function as early as 1834. The famous Bolzano-Weierstrass Theorem today cannot be discussed without a knowledge of limit points. Although Weierstrass never defined *limit point* as a concept, it was not uncommon in his time to work with a concept without having an explicit definition for it [12]. Thus there are several places where one can see the arguable beginnings of point-set topology. Cantor's 1872 paper [6], the main focus of this article, is chosen as our object of study because it begins with a very clear and well-designed problem in analysis and solves this problem by introducing the derived set, a purely point-set notion. Thus it builds a “motivational bridge” between familiar concepts in analysis to a new concept in point-set topology, addressing the problem of point-set topology being disconnected from other branches of mathematics.

This paper has a two-fold purpose. First, it introduces the reader to Cantor's 1872 paper and in particular, his need for a theory of "nearness without distance" in order to solve an analysis problem. Second, the paper provides information to the teacher who would like to introduce point-set topology in an historical context, motivated by some of the questions that were popular at the time. Our two objectives are combined in Sections 7 and 8, where we present Cantor's main theorem in a way that should be accessible and provide motivation for the study of point-set topology. We begin with a brief discussion of the mathematical climate at the time of Cantor's 1872 paper. We use modern notation and parlance to convey Cantor's ideas, whenever doing so would make it easier for the modern reader of mathematics to understand and when there is no possibility of losing any of Cantor's original meaning or intent. (*Editor's note:* All passages from Cantor's 1872 paper that appear in this article were translated from German to English by the author.)

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.

In order to modify the proof in his 1870 paper, Cantor needed to develop his own theory of the real numbers. Although our purpose in examining his theory of real numbers is to demonstrate the emergence of point-set topology, Cantor's construction is interesting in its own right. Even Bertrand Russell begrudges Cantor a compliment when he writes

The theory of Cantor . . . with all the requisite clearness, lends itself more easily to the interpretation which I advocate, and is specially designed to

provethe existence of limits. [22, p. 283, emphasis original]

Weierstrass apparently had a theory of the reals at this point ([10, footnote 47], [21]). However, none of Weierstrass' writing on the theory of reals seems to have survived, and we only know of it through his students. Since his students all seem to have published their teacher's theory after 1872, it may not have been well known when Cantor wrote his paper. In any case, in his 1872 paper, Cantor cites Euclid as the definitive source for the theory of real numbers, writing

For comparison ... we mention the book “Elements of Euclid” which remains the decisive treatment of the subject. [6, p. 127]

Still, Cantor was not fully satisfied with this theory, and he desired a more solid foundation for standard operations (addition, etc.) performed on the real numbers. To this end, he defined a sequence of rational numbers \(a_1,a_2,\dots,a_n,\dots \) to possess **numberness** (*Zahlengroße*) or to be a **fundamental sequence** if for every \(\epsilon > 0, \) there exists an integer \(N\) such that whenever \(n\ge N,\) \( \vert a_{n+m}- a_n\vert\,<\epsilon\) for any positive integer \( m.\) (In modern terminology, Cantor has defined a *Cauchy sequence* of rational numbers.) We associate to any rational sequence possessing numberness a symbol \(b,\) which we refer to as a **number value.** Cantor writes

This property of sequence \(a_n\) I express in the words \(a_n\) has a certain limit \(b.\)

In other words, by the symbol \(\lim a_n,\) Cantor means the number value \(b\) associated to that sequence \(a_n.\) He denotes the collection of all such fundamental sequences by \(B.\) We will see in Section 5 that every real number corresponds to a fundamental sequence of \(B.\) Cantor goes on to show how we may (for now at least) conceptualize \( B\) as the set of real numbers. However, this is only for intuitive purposes. Cantor is explicit about this last point when he writes

Now these words initially have no other meaning except as an expression for

thoseproperties of the sequence, and from the fact that we associate to the series \(a_1,a_2,\dots,a_n,\dots \) a special character \(b,\) it follows that with various series, various characters \( b, b^{\prime}, b^{\prime\prime},\dots\) are formed.

Immediately after making this definition, Cantor is quick to note that the number value \(b\) is simply a formal symbol associated to the sequence \(a_1,a_2,\dots,a_n,\dots .\) This is important to note, as Cantor did not wish to fall into the error of assuming the existence of limits of sequences of real numbers. He simply associates a symbol to any fundamental sequence. Next Cantor defines a total ordering on the set \(B.\) Let \( b, b^{\prime}\) be number values with corresponding fundamental sequences \(a_1,a_2,\dots,a_n,\dots \) and \({a_1^{\prime}},{a^{\prime}_2},\dots,{a^{\prime}_n}\dots ,\) respectively. Then one of the following three relations must hold:

- For every \(\epsilon > 0,\) there exists a positive integer \(N\) such that for every \(n\ge N, \) we have \({a_n}-{a^{\prime}_n} < \epsilon.\)
- There is a rational \(\epsilon > 0,\) and a positive integer \(N\) such that for every \(n\ge N, \) we have \({a_n}-{a^{\prime}_n} > \epsilon.\)
- There is a rational \(\epsilon > 0,\) and a positive integer \(N\) such that for every \(n\ge N, \) we have \({a_n}-{a^{\prime}_n} < -\epsilon.\)

In the first case, Cantor defines \( b= b^{\prime},\) in the second \( b> b^{\prime}\) and in the third \( b< b^{\prime}.\) It is important to again stress that Cantor is defining formal symbol relations. The symbol \(=\) is an equivalence relation on \(B\) (although Cantor does not use this terminology), and we continue to write \(B\) for the set of equivalence classes of number values under the relation \(=.\) Technically speaking, then, an element \(b\in B\) is an equivalence class of fundamental sequences.

Now Cantor is ready to define the operations of addition, subtraction, multiplication and division in \(B.\) For conceptual purposes, we may think of this as defining said operations on all real numbers, but technically speaking we are only formally symbol-pushing. Let the number values \( b, b^{\prime}, b^{\prime\prime}\) correspond to the fundamental sequences \[a_1,a_2,\dots, a_n,\dots\] \[{a^{\prime}_1},{a^{\prime}_2},\dots,{a^{\prime}_n},\dots\] \[{a^{\prime\prime}_1},{a^{\prime\prime}_2},\dots,{a^{\prime\prime}_n},\dots,\] respectively. If \[\lim\left(a_n\pm{a^{\prime}_n}-{a^{\prime\prime}_n}\right)=0,\] \[\lim\left(a_n\cdot{a^{\prime}_n}-{a^{\prime\prime}_n}\right)=0,\] \[\lim\left(\frac{a_n}{{a^{\prime}_n}}-{a^{\prime\prime}_n}\right)=0,\,\,\,{\rm for}\,\,\,{a^{\prime}_n} \not=0,\] then we write \( b\pm b^{\prime}=b^{\prime\prime},\) \( b\,b^{\prime}= b^{\prime\prime},\) and \({\frac{b}{b^{\prime}}} = b^{\prime\prime},\) respectively.

After defining operations on \(B,\) Cantor constructs the set \(C\) from \(B\) in much the same way he defined \(B\) from the rationals. That is, he considers all sequences \(b_1,b_2,\dots,b_n,\dots \) of \(B\) such that the limit of \(b_{n+m} -b_n\) equals \(0\) for some fixed value of \(m.\) We associate a symbol \(c\) to such a sequence and define relations, including \(=, <, >, \pm, \cdot,\) and \(÷,\) among such \(c\) as we did in \(B.\) Continuing in this manner, Cantor is able to construct the sets \(D,E,\dots\) consisting of equivalence classes of fundamental sequences made up of members of the previous set. Cantor then remarks that,

It is reserved for me to come back to all these conditions on another occasion in more detail,

a statement that, at least with hindsight, makes his construction here a precursor to his famous theory of transfinite numbers.

Cantor creates a bijection between the number line and number values. He calls an element on the number line a “point” and an element of \(B\) a “number value” or “value,” and seeks a bijection between values and points. Again, the idea here is that after we have established the bijection, we will then have a rigorous foundation to justify arithmetic operations on the real numbers.

Fix a point \(o\) on the number line (think of this as the origin). To show that the number line injects into the collection of number values, Cantor first considers the case when the distance from the given point to \(o\) has a rational relationship. If point \(a\) on the number line is rational, then we may associate the constant sequence \(a,a,\dots,a\dots,\) which possesses numberness, to the rational number \(a\) on the number line. This seemingly obvious fact is not without its criticisms. As Russell [17, p. 285] points out

There is absolutely nothing in the above definition of the real numbers [number values] to show that \(a\) is the real number defined by a fundamental series [sequence] whose terms are all equal to \(a.\) The only reason why this seems self-evident is, that the definition by limits is unconsciously present, making us think that, since \(a\) is plainly the limit of a series whose terms are all equal to \(a,\) therefore \(a\) must be the real number defined by such a series.

However, this may be problematic only when dealing with the arithmetic of number values, not when defining a bijection with the reals. So if point \(a\) is rational, we associate \(a\) with the sequence \(a,a,\dots,a,\dots\); that is, with whatever symbol the limit symbol of the sequence may be. Now suppose that a point \(b\) on the number line is irrational. Cantor asserts that

It is always possible to give a sequence \[a_1,a_2,\dots,a_n,\dots\quad\quad\quad\quad(1)\] [so that the] distance ... from the point \(o\) is equal to \(b,\) where \(b\) is the corresponding numerical quantity of sequence \((1)\).

In other words, Cantor associates the equivalence class [\(a_n\)] \(\in B\) to \(b.\)

Cantor took the converse, that "the geometry of the straight line is complete," as an axiom. He writes,

[T]o make the geometry of the straight line complete is only to add an axiom, which simply consists in [declaring that] any numerical quantity belongs to a certain point of the straight line ... I call this theorem an

axiombecause it is in its nature to not generally be provable. (emphasis original).

Before diving into what we now call topology, Cantor points out that this bijection is helpful only for conceptual purposes.

Given the previous, we now assign number values to the points on the line. For clarity (not that it is essential), we use this notion in the following and have, when we speak of points, values in mind by which they are given.

Notice that once again, Cantor stresses that thinking of points on the real line as number values is "for clarity" and not essential to what he is doing.

Though it may seem like the work we have done in Sections 4 and 5 is not at all related to topology, the building of such a rigorous foundation is necessary to make the definitions that appear in Section 6. We will see that these definitions abstract away the distance between real numbers, but keep the nearness between them, precisely by making the distinction between "points" and "values."

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

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

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 infinite 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.

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 intervalin 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.

**Conclusion **

Cantor's 1872 paper made an important contribution towards the development of point-set topology. His detailed and meticulous construction of the real numbers made it possible to build a rigorous foundation for what we might now refer to as "nearness without distance." These ideas can be used to place point-set topology within the larger mathematical world and motivate the study of topology via analysis. It is our hope that this beautiful branch of mathematics will not be lost on students because it appears to them to be totally removed from what they believe to be "real" mathematics.

**Acknowledgments **

The author would like to thank anonymous referees for their helpful suggestions and comments as well as *Loci: Convergence* editor Janet Beery for greatly improving the exposition of this paper.

**About the Author **

Nicholas Scoville earned his bachelor's degree in mathematics from Western Michigan University in 2003 and his Ph.D. in algebraic topology from Dartmouth College in 2010. He is assistant professor of mathematics at Ursinus College in Collegeville, Pennsylvania, where he advises summer research projects in discrete Morse theory as part of the NSF-funded Ursinus College Research Experience for Undergraduates (REU).

**Links to Related Resources**

**Point-Set Topology **

Topology Papers Project: Collection of historic papers important to the development of point-set topology, compiled by the author of the present article, Nicholas Scoville, Ursinus College

**Differential Equations**

ODE original sources bibliography containing many of the original papers in differential equations, compiled by Adam Parker, Wittenberg University

Related article: "Peano on Wronskians: A Translation," by Susannah Engdahl and Adam Parker, both of Wittenberg University, here in *Loci: Convergence*, is the result of a student-faculty collaboration. It offers a model for student translation projects and ideas for how to use the translation resulting from this particular collaboration in class.

**Discrete Mathematics**

Teaching Discrete Mathematics via Primary Historical Sources describes class projects from Phase I of this project of Jerry Lodder, David Pengelley, and colleagues, New Mexico State University. Phase II includes projects based on original sources for a variety of mathematics and computer science courses.

**Mathematics of Leonhard Euler **

The Euler Archive contains hundreds of Leonhard Euler's papers, many in English translation.

Related articles:

"Teaching and Research with Original Sources from the Euler Archive," by Dominic Klyve, Lee Stemkoski, and Erik Tou, here in *Loci: Convergence, *offers ideas for how to use Euler's papers with students.

"Investigating Euler’s Polyhedral Formula Using Original Sources," by Lee Stemkoski of Adelphi University, here in *Loci: Convergence, *offers more specific ideas on how to use Euler's papers with students.

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