Chapter 3

Chapter 3
The Formative Years

The department at Tennessee surely must have been considerably below his experience at Chicago and, indeed, Texas. There were only two faculty, Schmitt the chairman and Moore. The undergraduate offerings were no more sophisticated than calculus. The graduate offerings were described as follows in the 1906-1907 catalog:

1. [a.] Higher Algebra. In this course a knowledge of all the preceding courses is presupposed and the principles of algebra are viewed in their widest applications. Chrystal's Algebra and lectures. Senior and Graduate. Fall. Two hours.
2. [b.] Trigonometry of Imaginaires. A course embracing Demoivre's Theorem, hyperbolic trigonometry, and many other subjects not included in courses four and five. Lock 's Higher Trigonometry. Senior and Graduates. Winter. Two hours.
3. [c.] Modern Geometry. The principles of continuity, duality, inversion, harmonic ranges, poles and polars, and other subjects are considered. Taylor 's Euclid, V-VI. Senior and Graduate. Spring. Two hours.
4. []Should there be demand therefore, one or two of the following courses will be given in 1906-1907:
5. [d.] Foundations of Arithmetic and Geometry. Including a consideration of various sets of postulates and a study of relations existing between fundamental propositions. (In particular a discussion of the genesis of the ordinary number system from postulates concerning positive integers.) Prerequisite: A certain aptitude for close critical reasoning.
6. [e.] Non-Euclidean Geometrics including the Bolyai - Lobachevshian and Riemannian geometries.
7. [f.] Theory of Functions of a Real Variable, including a study of limits and continuity.
8. [g.] Theory of Functions of a Complex Variable. It is desirable that there should have been a previous careful study of limits and continuity.

No doubt Moore experimented with his teaching approach and this may not have been well received in all quarters. Moreover, there did occur a difference between Moore and the chairman. Following the end of the first semester Moore was slow to turn his grades in to the appropriate office. In fact, he had not even begun to grade the papers when he was approached on the campus one day by the chairman, Schmitt , who inquired, "Professor Moore, I notice that you have not yet turned in your grades. Have you finished grading the papers?" Moore replied that he had not, but did not elaborate further; that is, that he had not even begun. Sometime later the chairman became aware that, at their earlier meeting, Moore had not even begun grading the papers. He called Moore in and accused Moore of lying to him.⁶³ This, of course, was not received with the happiest of feelings by R. L. Moore and may have been a major factor in his decision to leave Tennessee. However, his decision to leave Tennessee may have come simply because he felt that there was not faculty or student potential enough to merit his sustained effort. It is not surprising that the offerings described in the 1906-1907 catalog, to be offered if interest developed, were not offered. Moore had left and the beginning of the 1906-1907 academic year found him at Princeton.

One other alteration had occurred during the year at Tennessee. Moore had grown a mustache, and though he shaved it before returning home, he did send home a picture. He felt his father was pleased by it; indeed, his father had been full-bearded as long as Robert Lee Moore could remember.

Halsted had been away from Texas several years by 1906, but he no doubt learned of Moore's move to his own alma mater. There is some reason to think that Halsted yearned to see a successful protege of his take a position at Princeton.

R. L. Moore had been preceded to Princeton by Oswald Veblen , who went there as Preceptor the same year Moore had gone to Tennessee. The preceptor system is attributed by some to Woodrow Wilson .

Our class was the one who had fortune to have the Preceptors, fifty of whom joined the teaching faculty. In some courses they met a small group of students, not more than ten, and we met in rooms on the campus or in their living quarters. We would discuss with our preceptors about our assignments. I think my class did much better with that situation than to be in larger rooms with a large number of students. Many of these preceptors were promoted and many of them were well know while many of them remained at Princeton. Some of the mathematics teachers - some of them arranged to have some of them attend a room in the evenings. I frequently consulted with them, so that I learned enough to pass any examination I took.⁶⁴ My four years at Princeton University were under Woodrow Wilson . My class was the first to receive and have four years of his Preceptoral system with which you are, no doubt, familiar. It has something of the Socratic system in it. The President brought 50 new men to Faculty recruited from all over the U. S. As Preceptors they also had classes. The Prof in each course had us in the classroom, but one hour a week was subtracted from hours of courses and given to a Preceptor who had no more than five men in his Study or room for that purpose in recitation building. In this informal atmosphere we discussed the course, were assigned some outside reading on which we were asked questions later. So we had two men for every course and it was helpful and stimulating. Many of the original 50 rose from the rank of Assistant Prof. to Heads of Departments and some to becoming Deans. Princeton still clings to this system. It required a larger Faculty and that is why Princeton with a smaller number of students than Yale and Harvard has a larger Faculty than either of them.⁶⁵

Wilson had attracted some fifty men to Princeton at the outset of his presidency at Princeton. They served as preceptors; that is, as individuals who taught classes and who also served as assistants in other classes. In those instances, a professor would have his usual weekly class time reduced by one hour. During that hour students of his class would meet with the preceptor in the preceptor's study or room to discuss the ideas of the course, or else to discuss readings which had been assigned earlier in the regular class. No preceptor was assigned more than five or ten students. Princeton thus early established a low student-faculty ratio and, to this day, has a sizeable faculty, for the number of students there, if compared to many other universities.

Thus it was that Oswald Veblen , and then a year later, Robert Lee Moore, went to Princeton as Preceptors. However, the Princeton system was not as individualized as might be suggested by the concept of the preceptor. Indeed, many of the courses were taught in more than one section, but were required to cover the same material. This requirement came by way of "departmental" examinations which were administered to each section, regardless of instructor. Moreover, the several instructors would collect together to grade the examinations en masse, with each grader having responsibility for specific problems. This surely reduced the effect of the day-to-day evaluation by the teacher. Moreover, the system probably caused some to teach "toward" the examination, instead of striving to develop the student's skills or abilities. Although R. L. Moore had held his doctoral degree for only one year, prior to coming to Princeton, this was not a philosophy consistent with his experience as a student of mathematics. Neither did it fit with his desires to teach that subject in such a fashion.

The attitudes of some in the mathematics faculty grated on him. In one instance, a large class was having a test administered and each student had been instructed beforehand to bring a "blue-book" instead of loose-leaf paper. After the test had begun, one of the faculty proctors stopped in front of the desk of a student, reached down to remove a blue-book and announced in a very loud voice, "Only one blue-book!" It was the sort of person who would do that sort of unnecessary thing that disturbed Moore.

It is not surprising that Moore sought association outside the faculty of the mathematics department. One of his most highly regarded associates was "Spider" Relley, the boxing instructor at Princeton. Moore was an enthusiast for competitive activity, though to interest him, it needed to be individual in nature. Boxing fit Moore's requirement exactly, and he took to that sport with remarkable dedication. Then, too, as he came to Princeton, he was recovering from malaria and physical exercise would tend to overcome the residue of his illness. He was never a large man, though he was quite active. Shortly after going to Princeton, his weight was recorded at 143, though he weighed as much as 165 before he left there. The boxing he did at Princeton was of a sparring nature, not professional, and with whomever might be at the gym and wanting to spar. The procedure was that two boxers would agree to spar and they would decide whether to "go hard" or "go easy." While they would not attempt to knock each other out, a decision to "go hard" simply meant that they would not pull their punches. Since they had no one to time their rounds, they would spar until one of them called "stop!" Moore never felt that he should call "stop" but rather, that he should let the other person decide when to stop. One time though, he very nearly called "stop" and was quite relieved when the other man did. He had no idea how long they had sparred, but finally he could not even hold his arms up to defend himself. He let his arms drop to his sides, and still felt that he might be able to deliver some kind of blow by swinging up. Right then, the other man said, "stop" and Moore found that a considerably pleasant sound. Most of the men, who Moore fought, were larger than he. One, in particular, was about 6'3" tall and, accordingly, had a substantial advantage in reach. This man would invariably want to "go easy," but in the midst of sparring, either man might say "go hard" and immediately change the level of aggression. However, this particular man would decide, without announcing it, to "go hard." Moore learned after awhile, to just "go hard" with this man.

Moore's recovery time from his bout with malaria must not have impeded his boxing at Princeton. At one time, not long after he had begun boxing there, Spider told him, "Well, I'll tell you what, Mr. Moore, you don't ever have to worry about having any bad effects from malaria. I'd hate to see you if you were a totally healthy person. I've never seen anyone who fights the way you fight."⁶⁶

Moore would literally run over to the gymnasium, when he got out of class at Princeton, so that he would have as much time boxing as possible. Once at the gymnasium, Moore would sit and wait until someone came in who wished to box. They would box awhile, until the opponent would decide to leave, and then, Moore would sit down to wait until someone else wished to box. After awhile that opponent would leave and Moore would again sit down to wait for someone else to come along. This would go on for two or three hours. At one point he was practicing a right cross. He wanted to perfect that blow and he was boxing with a man ho was not as capable a boxer as he. Although he had no desire to hurt his opponent, the man led with a left jab only to have Moore return a right cross over the jab, more sharply than Moore had intended. The novice went down, surprising Moore perhaps even more than his opponent. In any case, Moore apologized profusely, not meaning to have hurt the man.

In 1931-32, Moore was the first native American mathematician to tour on the American Mathematical Society Visiting Lectureship. On that program he visited Princeton and quickly made his way to see "Spider" Kelley . Moore had been perplexed for years at an incident which had occurred while he was boxing at Princeton. He had been boxing with a man named Red Carpenter (the man had bushy red hair) and Spider just stopped them. Moore had never understood why they had been asked to stop. They were just boxing and he thought nothing was wrong. In any case, Spider had said, "Stop!," and they had stopped. No explanation was ever given. Thus, as he found himself back at Princeton he went to see Spider. He walked into the gym as if he had not been gone at all and Spider looked at him and said, "Well, Mr. Moore, I'm surprised to see you here." Moore replied, "Why did you stop Red Carpenter and me from fighting that time?" Spider began to tell him only to be interrupted by someone else who had come up to them. The explanation was never finished and Moore never learned why Spider had stopped them.

Moore impressed others besides his sparring partners with his aggressiveness. Once Moore's officemate was overheard asking another faculty member, "Have you ever seen Moore go into action?" The other replied, "Well, once he grabbed a cane from my hand and nearly broke my thumb!" Moore's officemate then related that, "I was sitting in my office talking with Moore and began talking in a way I knew I shouldn't, and suddenly, Moore got up, I found myself on the floor, and Moore was walking out the door." Moore had wished to express disapproval of the manner in which the officemate was talking, but did not wish to exchange words or physically fight his officemate. So he just got up, picked the man up from his chair, and dropped him sprawling on the floor. By leaving, his protest as effective and could almost be assumed to be humorous. In any event, his officemate tended to watch his tongue more closely after that.

It is obvious that R. L. Moore matured much during those years immediately following his graduate study. He suffered poor health while at Tennessee, made little mathematical contributions while there, and left after only one year. Moving the next year to Princeton, he discovered the sport of boxing and seriously participated in it, both for reasons of health and because he fiercely loved the sort of aggressive individual competition boxing offered. Finding himself teaching in a setting in which he found little in common with teaching philosophies of others on the faculty, he produced little mathematics but developed strong physical and mental independence. Perhaps it is proper to say that his already strong independence of will was further developed while at Princeton.

The only paper to appear, as having been submitted by Moore while at Princeton, was entitled "Geometry-on which the sum of the angles of every triangle is two right angles" (Trans. Amer. Math. Soc., V. 8, 1907, p. 369-378). However, this paper had been presented to the Society on April 22, 1905, as part of a paper entitled "Sets of metrical hypotheses for geometry." Within this paper a footnoted comment illustrates Moore's concern for clarity of statements:

"Dem" means an indication of a demonstration. Like wise, "by theorem" does not necessarily mean that this is the only theorem used in the demonstration. Similar statements may be made about the use, in this paper, of the words "therefore," "hence," etc. The sign = means "is, or are, identical with."

One might state this a little more suggestively, if less accurately, as follows: "While the parallel postulate, III, and thus all of that part of Hilber tian Geometry which follows, without use of his 'Axioms of ARCHIMEDES' and 'Vollstandigkeit Axiom,' can not indeed be proved from his other postulates I, II, IV, with III replaced by S, nevertheless it can be shown that a space concerning which these postulates (I, II, IV, S) are valid must be, if not the whole, then at least a part, of a space in which III also is true."

This result has an interesting connection with our spatial experience. Statements have been made to the effect that, since no human instruments, however, delicate, can measure exactly enough to decide in every conceivable case whether the sum of the angles of a triangle is equal to two right angles (unless the difference between this sum and two right angles should exceed a certain minimum amount), it is therefore impossible to settle the question whether our space is Bolyai -Lobachevski an or Euclidean even though it be granted that it is one or the other.

A consideration of various sets of postulates for arithmetic and geometry, and a study of relations existing between fundamental propositions. The development of arithmetic and geometry from a set of postulates. Graduate course, second term, 3 hours a week. Dr. Moore.

Moore's strong feeling concerning the teaching methods employed at Princeton is indicated by an incident which occurred at the University of Texas over forty years after Moore left Princeton. Professor Lefschetz was an invited speaker at the University of Texas and had for many years been on the Princeton faculty, though he came there following Moore's departure. Moore normally did not attend such lectures but, to show courtesy to a Princeton mathematician of high reputation, Moore did attend Lefschetz ' lecture, taking pains though to sit toward the back of the room. At the end of Lefschetz ' talk, as questions were being raised. Lefschetz asked Moore's opinion of his comments. Moore had not gone to the talk seeking an argument, but as he later put it, Lefschetz looked up at him and said, "Well, do you agree with everything that I have said, Dr. Moore?" Moore, realizing that Lefschetz was not about to let him be at peace, stood up and asked, "Do you still teach algebra at Princeton the way you did when I was there?"

When Moore left Princeton at the end of the academic year 1907-1908, he moved to an instructorship at Northwestern University. He remained there for three years. In 1908 the mathematics faculty numbered seven, and stressed geometry at the advanced level.

In 1908-1909, Moore taught plane trigonometry and analytical geometry, algebra, solid geometry and plane trigonometry, differential and integral calculus, and was scheduled to teach ordinary differential equations though that course was not offered in 1908- 1909. This course assignment was repeated the next year, including the offering of ordinary differential equations. However, by 1910- 1911 a new course had been added, "Non-Euclidean geometry," taught by Moore on a "to be arranged basis." With the departure of Moore from the faculty the next year, so departed that non-Euclidean offering from the mathematics catalog.

Moore did find his way back to Texas during his stay at Northwestern, at least to the extent that he found his way to Brenham, Texas on August 19, 1910 to be married to Margaret McClellan Key . The ceremony was performed at her family home with a Dallas pastor officiating.

Thus, when R. L. Moore moved from Northwestern University to the University of Pennsylvania in 1911, he brought to Pennsylvania his Texas bride, but no additional mathematics in published form. It soon was to become apparent, though, that much work had been done after his graduation from the University of Chicago, for Moore began to earn a considerable reputation for himself as a research mathematician.

The mathematician hired by the University of Pennsylvania named Robert Lee Moore was not yet thirty years of age. He had a sound academic reputation, though he had not blossomed forth with immediate substantive mathematics following his graduation from the University of Chicago. However, with the exception of Princeton, the University of Pennsylvania offered Moore a setting in which to function which was the most stable and substantial that he had found since leaving Chicago.

The chairman of the department at Pennsylvania was named Fisher who was for many years of some reputation because of a calculus text which he authored. Those of professor rank included , Schwatt , and Hallett .

The faculty at Pennsylvania was not without strength, in mathematics as well as individual strength of personality. Foremost among those who were of colorful personalities was Schwatt , an analyst who specialized in real variables and infinite series. Schwatt was not always kind in his remarks to students. As one of them recalls, "Schwatt kept saying 'Don't you understand? Mr. Smith , there's three kinds of fools (with a heavy accent) . . . there's just ordinary fools, there's damn fools, and then, there's you."'

Schwatt was of short stature, with a long nose, and a beard which students considered funny. He was a proselytizer of students and sought others to share his enthusiasm for series, particularly divergent series. He would chew tobacco (and spit) in the classroom. Occasionally this would be met with protest from a female student who would have backbone enough and a fracas would result. At one point Schwatt had the son of Bell , the Chairman of the Board of Trustees of the University of Pennsylvania, in one of his undergraduate classes. Schwatt , in reprimanding Bell 's son, is reported to have said, "Mr. Bell , there are fools, and damn fools and godamned fools. You belong to the third category!"⁶⁷

Along with Schwatt and Fisher , Pennsylvania also had as professors and Hallett . specialized in analytic and differential geometry and Hallett was an algebraist. Additionally, in 1911-1912 there were five other faculty holding rank assistant professor. The course offerings were broad, covering analysis, algebra, differential equations, geometry, and applied mathematics. By 1912 the chairmanship of the department had shifted to , with Professors Fisher , Schwatt , and Hallett , and Assistant Professors Safford , Glenn , Chambers , and Babb . Instructors were Mitchell , Moore, and Beal . Moore offered a course entitled Foundation of Mathematics, described in the catalog as:

The theory of positive integers as a basis for analysis. Rigid motion and correspondence with a number manifold as factors in determining the properties of space. Metrical and non-metrical spaces. A critical study of interrelations between different systems of axioms.

By the following year Moore had introduced a course entitled Theory of Points Sets:

Theory of sets of points in metrical and in non-metrical spaces. Contributions of Frechet and others to the foundations of point set theory. Content and measure. Jordan curve theory and other applications.

These two courses Moore offered throughout his tenure at the University of Pennsylvania, usually offering each course every other year, alternating the two of them.

R. L. Moore was first appointed as Instructor of Mathematics and he was to hold that rank until 1916. The mathematical period of dormancy, though, had passed for R. L. Moore. He was to have five of his papers appear in published form during the years 1912 through 1915, and two others were published in 1916. His work during that period dealt with analysis as well as continue effort toward Veblen 's axioms for geometry: Indeed, in 1912 he published "A note concerning Veblen 's axioms for geometry" (Trans. Amer. Math. Soc. 13 (1912) 74-76) and "On Duhamel 's theorem" (Ann. of Math. 13 (1912) 161-166). The first had been presented to the Society, in somewhat different form, on October 26, 1907 so it is clear that the mathematics in that paper had been completed before his move to Pennsylvania. Moore had offered some improvement of Veblen 's axioms by offering axioms alternative to Veblen 's, thereby reducing Veblen 's axioms by one. In the paper entitled "On Duhamel 's theorem," R. L. Moore dealt with a problem of analysis, using analytic methods. Duhamel had posed the proposition:

Let a_l + a₂ + ... + a be a sum of positive infinitesimals which approaches a limit when n = ¥. Let b_l + b₂ + ... + b be a second sum of infinitesimals which differ from the infinitesimals of the first sum by infinitesimals of higher order; i.e., let

lim ai bi = 1.

Then the second sum approaches a limit when n = , and this limit is the same as that of the first sum:

lim n = ¥  bl + b2 + ... + b = lim n = ¥  al + a2 + ... + a.68

Osgood had formulated an alternative theorem which was without some of the difficulties of application encountered by use of Duhmel's theorem. Then Moore proceeded to offer another version of the theorem "which is suitable for application to a still wider range of problems and at the same time seems to be even easier to apply."⁶⁹

By 1915 R. L. Moore was devoting more attention to "the linear continuum." On April 24, 1915 he presented a paper "On the linear continuum" to the American Mathematical Society, in which he refers to a set G of eight axioms which he had earlier proposed (The linear continuum in terms of point and limit, Ann. of Math. s. 2, v. 16, 1915, p. 123-133) for the linear continuum in terms of "point" and "limit." In that paper he had defined "betweenness" and had established that the set G is categorical with respect to "point" and the implied notion of "betweenness." He offered a new axiom, replacing Axiom 5 of the original set so as to gain a system which was "absolutely categorical," in the sense of Veblen (A system of axioms for geometry, Trans. Amer. Math. Soc. v. 5, 1904, p. 343-384).

It was in 1915 that R. L. Moore published a result which gained him the attention of the mathematical community. Moore and Veblen had both treated geometric problems in their dissertations. However, Veblen 's work seemed to be taking him toward a notion of non-metric considerations while Moore's work seemed to be taking direction toward the metric situation. In fact, Veblen had published in 1905 a paper entitled "Theory on plane curves in non-metrical analysis situs" (Trans. Amer. Math. Soc. 6 (1905) 83-98) in which he argued a proof of the fundamental theorem proposed by Jordan : A simple closed curve lying wholly in a plane decomposes the plane into an inside and an outside region. The setting for this theorem was taken to be a space satisfying Axioms I - VIII, XI of his dissertation, stating explicitly that nothing is assumed "about analytic geometry, the parallel axiom, congruence relations, nor the existence of points outside a plane." The results published in 1915 by Moore were rather devastating, inasmuch as he established that any plane satisfying Veblen 's Axioms I - VIII, XI is a number plane. That is, any plane, satisfying those axioms, contains a system of continuous curves such that, considering those curves as straight lines, the plane is an ordinary Euclidean plane. Thus, Moore claimed that any discussions of analysis situs based on Veblen 's axioms (as for example, Veblen 's proof of the theorem that a Jordan curve divides its plane into just two parts) is no more general than one based on analytic hypothesis.

The results obtained by Moore were remarkable and unexpected by many. The mathematics involved was difficult, in that Moore established a one-to-one reciprocal continuous correspondence between the continuous curves of Veblen 's planes and straight line of an ordinary Euclidean plane. If Robert Lee Moore had not a substantial reputation before publishing those results, he surely had established himself after those result were known. Moore's attention to and interest in geometry and particularly Veblen 's axiomatic approach had been consistent across the years since he departed from Chicago. In 1913 he presented a paper before the American Mathematical Society, "Concerning a non-metrical pseudo-Archimedean axiom," in which he had further investigated Hilber t's plane axioms of Groups I and II⁷⁰ and Veblen 's Axioms I - VIII.

It was after his remarkable result which showed Veblen 's axioms really gave rise to a metric plane, that Moore began to offer theorems which were not truly set in analysis nor were they straightforward investigations of axioms of geometry. In his paper, "On the foundations of plane analysis situs," (Proc. Nat. Acad. Sci. 2, 1916, pp. 270-272) he put forth the fundamental notions which were to form a basis for later development in point set topology. Indeed, in that paper, Moore stated:

The notion point, line, plane, order, and congruence are fundamental in Euclidean geometry. Point, line and order (on a line) are fundamental in descriptive geometry. Point, limit-point and regions (of certain types) are fundamental in analysis situs. It seems desirable that each of these doctrines should be founded on (developed from) a set of postulates (axioms) concerning notions that are fundamental for that particular doctrine. Euclidean geometry and descriptive geometry have been so developed. The present paper contains two systems of axioms, S₂ and, S₃ each of which is sufficient for a considerable body of theorems in the domain of plane analysis situs. The axioms of each system are stated in terms of a class, S, of elements called points and a class of sub-classes of S called regions.

Within that paper he stated the following definitions and axioms. These were to be the forerunners of remarkable developments in point set topology for at least the next half century.

Definitions

• A point P is said to be a limit point of a point-set M if, and only if, every region that contains P contains at least one point of M distinct from P.
• The boundary of a point-set M is the set of of all points [X] such that every region that contains X contains at least one point of M and at least one point that does not belong to M.
• If M is a set of points, M¢ denotes the point-set composed of M plus its boundary.
• A set of points K is said to be bounded if there exists a finite set of regions R₁,R₂,R₃... R_n such that K is a subset of (R₁+R₂+R₃+... +R_n)'.
• If R is a region the point-set S-R¢ is called the exterior of R.
• A set of points is said to be connected if however it be divided into two mutually exclusive sub-sets, one of them contains a limit point of the other one.

The System S₂

1. [Axiom 1.] There exists an infinite sequence of regions, K₁,K₂,K₃... such that

   1. if m is an integer and P is a point, there exists an integer n greater than m, such that K_n contains P,
   2. if P and [`P] are distinct points of a region R, then there exists an integer d such that if n > d and K<sub>n</sub> contains P then K¢ is a subset of R-[`P].

2. [Axiom 2.] Every region is a connected set of points.
3. [Axiom 3.] If R is a region, S-R¢ is a connected set of points.
4. [Axiom 4.] Every infinite set of points lying in a region has at least one limit point.
5. [Axiom 5.] There exists an infinite set of points that has no limit point.
6. [Axiom 6'.] If R is a region and AB is an arc such that AB-A is a subset of R then (R+A) -AB is a connected set of points.
7. [Axiom 7'.] Every boundary point of a region is a limit point of the exterior of that region.
8. [Axiom 8.] Every simple closed curve is the boundary of at least one region.

The System S₃

The system S₃ is composed of Axioms 1', 2', 3, 4', 5, 6', 7,' and 8, where Axioms 1', 2', and 4' are as follows:

1. [Axiom 1'.] If P is a point, there exists an infinite sequence of regions, R₁,R₂,R₃... such that

   1. P is the only point they have in common,
   2. for every n R_n+1 is a proper subset of R_n,
   3. if R is a region containing P then there exists n such that R¢ is a subset of T.

2. [Axiom 2'.] If A and B are two distinct points of a region R then there exists, in R, at least one simple continuous arc from A to B.
3. [Axiom 4'.] If R is a region, R¢ possesses the Heine-Borel property.

An example of a system satisfying S₂ is obtained if in ordinary Euclidean space of two dimensions, the term region is applied to every bounded connected set of points M, of connected exterior, such that every point of M is in the interior of some triangle that lies wholly in M.

At about this time, on April 6, 1915, Dean Arthur Hobson Quinn wrote, as chairman of the College of Arts and Sciences nominating committee, to Dr. Josiah H. Penniman , Vice-Provost, "The Department of Mathematics recommends the promotion of Dr. Robert Lee Moore, Instructor in Mathematics, to the position of Assistant Professor of Mathematics . . . " On May 4, 1915 Dean Quinn wrote Dr. Edgar F. Smith , Provost, a letter in which the same statement was made. The promotion was approved and the academic year 1916-1917 saw Robert Lee Moore at the University of Pennsylvania holding rank Assistant Professor.

The year 1916 was one in which R. L. Moore graduated his first doctoral student. J. R. Kline graduated that year, having first come to the University of Pennsylvania in 1912 after graduation from Muhlenberg College with an A. B. degree. He received his M. A. from Pennsylvania in 1914 and his Ph.D. in 1916. His formal courses, with fellow students, under Moore, were not many. Moore alternated from one year to the next his courses entitled "Foundations of Mathematics," and "Theory of Sets." Beyond those, individual study was the fashion, with Moore encouraging some to work with him while discouraging others. Kline was able to establish the following converse to the Jordan theorem:

Suppose K is a closed set of points and that S-K = S₁ + S₂, where S₁ and S₂ are non-compact point sets such that

1. every two points of S_i (i = 1, 2) can be joined by an arc lying entirely in S_i
2. every arc joining a point of S to a point of S contains a point of K
3. if 0 is a point of K and P is a point not belonging to K, then P can be joined to 0 by an arc having no points except 0 common with K. Every point set K that satisfies these conditions is an open curve.⁷¹

By 1918 Moore had further developed his mathematics and his teaching techniques. Graduating in 1918 was G. H. Hallett , a son of Professor Hallett of the mathematics faculty. An idea of the manner of teaching used by R. L. Moore is given by Hallett 's description:

He taught in a very remarkable way. He didn't give us any books. We didn't consult books at all in that course. It was a course in point set theory and he gave us certain axioms to start with and then we were asked at the beginning to prove certain theorems that we were told were true, given those axioms. We would work on the proofs and come back into class and he would ask how many people had the proofs and those who said yes were given a chance, at least one or two of them to give their proofs. And the other members of the class listened carefully to see if they made any mistakes and if a member of the class thought so, he would speak up and say why. And quite often, there were mistakes in the proof that were caught by the class and sometimes, nobody had the proof the first time and he would let us have another week at that one. Ordinarily, we would come in with the proofs. As the course went on though, some of the things given us were more difficult. He would give us a problem and ask what the solution to the problem was without telling us the theorem that was supposed to result. Then we would work on that. I remember that one of the theorems that I proved, he said had never been proved until the year before, and I had given him a different proof. I think, I'm not sure, he may have published that, but that's one case of that kind that I remember. He gave us a problem once which we worked on and which none of us got. He gave us a couple of weeks to work on it and none of us got it. And he said, "Well, I guess you needn't spend any more time on that. This is a problem mathematicians have been working on for centuries and nobody has ever solved it. I just thought you might, just by accident, be able to do something.⁷²

It well could be that Moore's manner of teaching influenced others. Hallett describes the teaching style of Mitchell of the mathematics faculty, reminding one at once of Halsted 's style of teaching:

One other course I took at the same time which was somewhat similar. It was taught by Professor Mitchell whom I also liked very much. He took a book, I think it was by Dr. Pierpont at Yale. I think it was in the area of functions of a real variable. I guess Professor Mitchell had found on inspection that not all of Professor Pierpont's proofs held up, so the way he taught this course in that subject, he gave us this book, but asked us to go through all the proofs that were given and find out whether they were watertight proofs or not and if not, why not. So time after time, we would find that they weren't and it almost always occurred when the author said, "Now, it is evident that . . . " And we would inspect and we would find quite often that it wasn't even so. It seemed plausible that it wasn't so. Actually this course had many of the elements of the other course. We'd discuss the matters in class together⁷³

Moore also had incorporated some of Halsted 's teaching style in his own approach to the classroom activity. Halsted often had been known to discuss matters foreign to mathematics in his classes. It was toward the end of World War I that Hallett had a course with Moore and recalls:

. . . that was during the last days of the first world war and Dr. Moore would frequently come in the class and take five minutes discussing something that had just appeared in the papers about the arguments of the logic, or lack of logic, in some statements by an important public figure connected with the war or other public events at that time. He and I were always getting a good deal of satisfaction in picking out the lack of logic in a good many of these statements by important public figures. That was sort of a forerunner to applying the same methods of thinking to public affairs which I went on with after graduation and for the rest of my areer. I think this added to the interest of the class and probably made it more attractive. And I think it was really of great value to point out that the same sort of criteria could be applied widely in fields where we weren't just dealing with mathematics.

Hallett continued:

I think I should make the observation that Dr. Moore's method of teaching brought out what appeared to me to be the two most important faculties in mathematical research - one which would surprise most people, I'd regard as imagination and the second, the ability to critical analysis in applying logic to what you think of to try out. And this same criteria, of course, apply to almost everything else. It is a method of thinking. And I think such success as I've had in the work I've done in the field of government probably has a good deal to do with that - because they don't catch me up very often in theories of logic in bills, or different parts of bills, that don't hang together.⁷⁴

Hallett , upon graduation, was offered a job as a secretary of the Proportional Representation League. He rejected an offer of a position at Rice Institute to accept a post with the League and went on to an illustrious career in public service.

While Moore was at the University of Pennsylvania the Department of Mathematics was housed in College Hall, the original building in West Philadelphia for the University of Pennsylvania. It was a stone building and Moore was officed in the basement in a fairly large room, with seven or eight faculty of instructor or assistant professor rank. It was right next door to a stable where they kept the horse that pulled the lawn mower over the campus. It was not a very pleasant place for faculty offices.

At least one of Moore's students from Pennsylvania likened him to a football coach, "He was always recruiting people."⁷⁵ One of the students he recruited was Anna Mullikin , who had come to the University of Pennsylvania from Goucher College. Moore did not hesitate to recruit a female student, so he quickly introduced her to his style of teaching. In some classes other students felt the classroom activity omitted them, with most of the conversation occurring between Moore and Mullikin . Anna Mullikin 's experience with Moore was similar to that of Hallett . She, too, was impressed with his unusual method, recalling:

He had his work all published, you see, and people would go and look it up in the library and he didn't want them to do that. He wanted them to work it out themselves. And he put them out of class if he found out that they were cheating. (In one class) there were three of us, but he put everybody out but me. One was a Catholic nun and she tried to get help from me and he put her out. He said if she needed help she didn't belong in his class.⁷⁶

By the post World War I years, Moore had developed his fundamental approach to teaching. He was to modify it, depending on the setting and the level of students, but the foundations of the method were not to alter. He would pose problems for the students and demand that they attack those problems on their own initiative and ability, resorting not at all to source books or other people. He was with extreme patience toward those students who would struggle through efforts at proofs, but with extreme impatience toward those who would violate the rules of study which he laid before them. He simply and immediately dismissed them from class.

Though he found support for his approach among some, there were those students who would not tolerate such a style and faculty who opposed such a method. Moore's method of teaching graduate mathematics offered contrast with the catalog description of requirements for a Ph.D. in 1913-1914:

Students were especially advised not to undertake an unduly large number of lecture courses at one time. The place for graduate study is the library rather than the lecture room. The most that can be done in the lecture is to guide the student into a general acquaintance with the principles of a subject, and it remains for him to broaden his knowledge and develop the details by extensive reading and private study.

Moore's personal values were not to be tampered with. He felt strongly about his southern heritage, honor, and gentlemanliness. One of his office mates was named Beal , a pleasant person who was slightly crippled. He and Moore were good friends, often lunching together. Moore seemed to some, including Beal , to prefer having girls in his classes. Perhaps he felt they were easier to draw out. Once Beel began kidding Moore over his preference for female students and Moore took offense. He didn't wish to hit Beal , but wished to make his protest effective. So, he quickly arose, picked Beal up from his chair and dropped him on the floor as he left the room. By taking that tack, the incident could be considered as humorous, but the point was well made. Later, someone reportedly asked Moore, "Well, what about that? Why did you do such a thing?" And Moore said, "Well, I couldn't hit a cripple!"⁷⁷

Moore did not successfully recruit all those students he sought. One young lady, Clara Williams on, had gained entry into the University of Pennsylvania, though it was predominantly a male school. She took calculus and differential equations from Moore and then was encouraged by Moore to study for her doctorate with him. She decided against such an effort, even though he had invited her to work toward her Ph.D. with him.

While Anna Mullikin was involved in her dissertation effort, in 1919, Moore came into his office in College Hall in a state of some excitement and sat down in a chair by the desk of a young instructor, Joseph Thomas . The only telephone in the room was on that desk and Moore evidently had something important on his mind. Thomas offered to leave, being the only other person in the office at that time. Moore said, "Oh, no, no, it doesn't make any difference," and he called , the chairman of the department. Moore informed that he had received an offer of an Associate Professorship from the University of Texas and made an appointment to see about it. The outcome was that R. L. Moore chose to return to Texas, accepting the Associate Professorship at his alma mater. At that time, he was an established mature mathematician, who was beginning to have success in producing students. He was 37 years of age and had been at Pennsylvania for almost ten years. He had one doctoral student in progress and she was to follow him to Texas and complete her dissertation there.

Halsted had been gone from Texas for years, having left there in 1902. But M. B. Porter , Halsted 's student who had achieved early success and who was on the faculty at Texas while Moore was an undergraduate, had gone away and now had returned to the University of Texas as departmental chairman. The idea of returning to Texas had natural appeal for Moore, as it often has for many Texans who find themselves on "foreign soil." He had some knowledge of Porter and an idea as to how he could function in a department chaired by Porter . Then, too, Mrs. Moore had not been in good health. She had suffered a severe fall in a subway and Moore had been careful to nurture her health. Perhaps the warmer climate of Texas was inviting to them both. In any case, the academic year 1920-1921 found R. L. Moore no longer at the University of Pennsylvania.

During Moore's tenure at Pennsylvania, he evolved from a promising mathematician to one of stature. He began serving as an Associate Editor of the Transactions of the American Mathematical Society in 1913 and continued in that capacity until he left Pennsylvania. He was a member of the Council of the American Mathematical Society in 1917-1919. He had produced substantial mathematics during his years at Pennsylvania and had begun the developments away from pure geometry toward those concepts which would become known as point set topology. He produced two doctoral students prior to leaving Pennsylvania (Kline and Hallett ) and had another in progress (Mullikin ) who would formally receive her degree from the University of Pennsylvania, though completing her dissertation while at Texas. Of his three doctoral students, each would make substantial contribution to his chosen profession, though the three would work in decidely different settings. Moreover, each would credit Moore's influence even though his student-teacher contact was brief. Kline would establish himself well as a university academician. Mullikin would have a long, worthwhile career as a high school teacher of mathematics. Hallett would experience an illustrious career in public service. Each would feel that Moore's style of teaching gave far more than just information to them; it developed their power of rational thought and this would stand them in good stead throughout their careers.

The University of Texas, with the hiring of R. L. Moore, had added to its faculty an established teacher, researcher, and producer of doctoral graduates. At the time of his first appointment, his research already accomplished gave greatest support to his professional reputation. He was gaining acclaim as a teacher. In fact, he had influenced E. H. Moore to teach in the manner which R. L. Moore had developed. While visiting at the University of Chicago one summer, he had lunched one day with E. H. Moore and L. E. Dickson . A topic of the conversation then between them dealt with effective methods of teaching mathematics. R. L. Moore explained his method, describing the procedure of posing questions or theorems for students and insisting that the students settle those questions on their own, without assistance of any sort save their own capability. Dickson tended to quickly deride that approach, but E. H. Moore , as was his wont, said little. He customarily gave some thought to new ideas before reacting to them. Later, E. H. Moore told R. L. Moore that he felt that approach was with much merit, and began employing it in his own classes. Even though he was known as a capable researcher and gaining stature as a teacher, perhaps no one at all predicted the extreme success which was to be R. L. Moore's at Texas. It was to be almost as if he had left Texas, spent time wandering about preparing himself until both he and his method were seasoned and mature. Then he returned to Texas to wring remarkable results from a raw and rather undeveloped setting.