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R. L. Moore

The Beginning of Topology in the United States and The Moore School1
F. Burton Jones


Jones, F. Burton. "The Beginning of Topology in the United States and The Moore School," in C.E. Aull & R. Lowen eds., Handbook of the History of General Topology, Volume 1. Kluwer Academic Publishers: 97-103 (1997).

After a gestation period of about ten years (1905-1915), topology began to be thought of as a coherent collection of mathematical topics mainly useful to analysis but of sufficient interest in themselves to stand alone (Hobson and W.H. Young). Courses were formulated which restricted the subject matter covered to Topology (or Analysis Situs as it was frequently called). Mostly such courses were designed to develop the student's knowledge and research ability so that he could do the work for the Ph.D. degree. Although Lefschets' "Topology" and R.L. Moore's "Foundations of pointset theory" appeared in the AMS Colloquium Publications as Volumes 12 and 13 in 1932, they were not textbooks. (In fact, writing textbooks was not the thing to do if one were to be taken seriously as a researcher and usually there wasn't the time). While definitely limited, these books were of considerable help in formulating the knowledge and tools necessary for research. (Even Schoenflies: "Entwickelung der Lehre von den Punktmannigfaltigkeiten" if one avoided or learned from his mistakes, was useful, as was Hausdorff's "Mengenlehre".) The first book that found this niche and became required of all (and the "bible" of some) was the first 100 pages of Alexandroff-Hopf's "Topologie" (1935) and a little later much of Kuratowski's "Topologie I and II". And finally John Kelley's "General Topology" in 1955 became "Topology that every analyst ought to know" and for the topology major the main source of his strength. By now undergraduate courses were being designed to prepare students for courses in analysis and "in Kelley".

What made this development so rapid and successful (as indeed it was) must in no small measure be assigned to E.H. Moore, and the group of young, vigorous and talented graduate students attracted to his young and outstanding department at Chicago. Stimulated by the faculty and each other, research in topology was established as respectable and rewarding. By 1920 this group seemed to have been broken up and scattered all over the nation: Birkhoff to Harvard, Veblen to Princeton, R.L. Moore to Penn (and Texas), Chittenden to Iowa, are easy to recall. Where each of them landed the process was repeated. Perhaps it is too much to compare anyone of them with E.H. Moore? But each of these was no ordinary mathematician. Birkhoff discovered that a measure preserving homeomorphism of a plane annulus onto itself which moves the points of the inner circle one way and points of the outer circle the other way must leave at least two points unmoved (i.e., it has to have at least two fixed points). This is a fantastic creation of the mind; and that mind must have some power to work out a proof. Chittenden in 1917 proved that if a space has a regular écarte (i.e., a distance function which makes the triangle inequality approximately true in the small) then the space is metric (it has a distance function for which the triangle inequality is true). At that time many (almost all) topologists including its originator Fréchet tried to solve this problem and failed. In 1916 R.L. Moore gave a topological characterization of the Euclidean plane. (More about this later.) These were men who produced lasting results.

But probably of more importance, these men produced mathematicians whose work added substantially to the scope and knowledge of topology (Whitney, Montgomery, Wilder, to name only a few). It was this sort of exponential growth that established topology as an essential part of the university curriculum so quickly. Of course the quality and usefulness of these results played a part in this development. But there was something entirely different, a sort of historical accident that was the most compelling--at least in set theoretic topology.

Toward the end of the last century interest in the foundations of plane geometry began to grow, so that by the turn of the century it is fair to say that the topic occupied some of the best minds. Hilbert in 1902 published a little book called "Foundations of Geometry" which became the focal point of much research. R.L. Moore was at that time an undergraduate student at the University of Texas. He was able to show that a part of one of Hilbert's axioms could be proved from the others, thereby shortening the set of axioms. E.H. Moore and Veblen were working in this area and invited R.L. Moore to come to Chicago for graduate study. Fréchet had at about this same time laid out in broad form some of the principle notions of topology and Lennes at Chicago gave, as it turned out, the correct definition of connectedness. Parallel to this was Canter's "Theory of Sets", controversy and all. So R.L. Moore was plunged into this caldron of ideas at a time when his imagination and power were strong--possibly at a peak. The challenge was clear. He wanted to characterize the plane in topological terms: point, region, limit point, connectedness, and prove from axioms given in these terms that the space possessed a system of curves that could serve as the lines of a coordinate system for the plane. In his thinking "point" and "region" were undefined terms having the properties laid down in the axioms. All other terms were defined in terms of "point" and "region". If you are not too touchy about sets, X denotes the set of all "points" and the set of all "regions" is a topological basis for the topology U of X. So Moore will try to find a few simple properties (axioms) which if possessed by X makes X homeomorphic to R2. I, his 1916 Transactions paper he states eight properties, Axioms 1 to 8. Axioms 2 to 5 are simple. Ax.2 says: Every region is a connected set of points. We would say X is locally connected. Ax.3says: If R id a region X - R is connected. We would say: X is co-connected. Ax.4 says: If R is a region, R has the Heine-Borel Property. We would say: X is locally compact. And Ax.5 says: There exists an infinite set of points that has no limit point. We would say: X is not compact.

Axiom 1. There exists a [countably] infinite sequence of regions K1, K2, K3,. . . such that (1) if m is an integer and p is a point, there is an integer n > m such that Kn contains p and (2) if p and q are distinct points of a region R, there is an integer l such that if n > l and Kn contains p then Kn and its boundary are a subset of R - q.

At first blush one sees that (1) X has a countable basis, (2) X is Hausdorff and (3) if H1, H2, H3, ... is an infinite subsequence of K1, K2, K3,. . . and p is in each of them, then p is in each of their closures and is the only such point. But here is a subtle sort of uniformity that Moore attributes to Veblen (AMS Trans. 1905). Not only is there a region containing p with a subscript arbitrarily large whose closure misses q and X - R but so do all those of larger subscript which contain p.

With just these five axioms Moore proves all the theorems 1 to 52. In fact in his beginning topology course this is what the students were required to do in the years before 1931. (Of course new material was added right along as it was discovered. Later it was from his 1932 book.) Listening to some of them talk about their student days it was Theorem 15 that impressed Moore. If you proved Th. 15 you were possibly Ph.D. calibre--if not . . .? This arcwise connectivity theorem is the tool that makes it possible to construct the curves that help establish the fact that X is a Euclidian plane (using all axioms 1-8).

Theorem 15. If a and b are distinct points of a connected open set M of X there exists in M a simple continuous are from a to b.

Axiom 1 implies that X is separable (some countable subset of X is dense in X). However, Moore remarks that when Axioms 1 and 2 are replaced by Axiom 1' (for each point p of X there is a countable sequence K1, K2, K3, . . . of regions that close down on p) and Axiom 2' (Every region is arcwise connected) all of the theorems 1-52 hold true even though the space need not be metric or separable.

Coming home from the "Boulder meeting" in the summer of 1929, Moore discovered his Automobile Road Space. Imagine that the roads are 30 feet wide and that there is one mile between adjacent junctions. One should visualize it as being locally like it would be out on the prairie. And as you get out toward the edge of the plane you may have to stretch the part outside of the roads to make room for the next sections. For there are no loops or undesirable intersections and each road is extended indefinitely straight away from the edge of the plane. (The "boundaries" of the roads do not belong to the space composed of these roads.) Regions are bounded by circles or simple closed curves for points on the edge of the plane. The sections of roads beyond the plane form an uncountable collection of disjoint connected open sets; hence the space consisting of all the roads is not separable. But the roads in the plane plus the edge points of the plane belonging to the roads is a separable subspace. The midpoints of the roads at the edge of the plane form an uncountable set having no limit point. So this subspace is not metric; hence the whole space is nonmetric (or nonmetrizable as some prefer).

roadspace image

The Automobile Road Space is an example of a nonseparable complete Moore space which is a 2-manifold. If one takes this subspace and identifies each pair of points of the 30 foot open interval symmetric to its center (and do this for each such interval where the roads leave the plane), one gets a separable nonmetric complete Moore 2-manifold. This process sort of folds each of these intervals up onto itself about its center.

By substituting Axioms 1' and 2' one avoids the trap that Veblen fell into in his paper: "Theory of plane curves in non-metrical analysis situs" AMS Transactions Vol 6 (1905). (Moore showed that none of the spaces in this paper could be nonmetrical.) And while there was considerable interest in generalizing metric topology in this way, was it worth losing Theorem 15 as a real mountain to climb and real test of research ability? (It is too easy to prove Theorem 15 on the basis of Axioms 1', 2', 3, 4, and 5.) Also it would be nice not to assume the existence of compactness, as in Axiom 4. Moore suggests this by not using Axiom 4 for Theorems 1-10. So with these compelling motives and guided by the Automobile Road Space example, Axiom 1 evolved into the Axiom 1 of his colloquium book of 1932. In this version the sequence of regions is replaced by a sequence of collections of regions each of which covers the space and contains the next (is monotone decreasing). While the sequence is countable as in the old Axiom 1, the collections need not be countable. After seeing what was required for the proof of old Theorem 15 (arcwise connectivity), a sort of "completeness" is tacked onto the end of old Axiom 1 (if a sequence of regions, one from each collection, is nested, then their closures have at least one point in common). Other parts of the axiom tell you not more than one point. (This is the way I remember it. The book seems to differ. Does it?) Old Theorem 15 becomes Theorem 1 in Chapter II when Axiom 2 (local connectivity) is added. There are 121 theorems in Chapter I and only a few require the completeness part. (Old Axiom 1 implies the new Axiom 1 if the completeness part is omitted). Remember that 52 theorems were proved on the basis of (old) Axioms 1 to 5. Axiom 2 remains the same: Space is locally connected. Axioms 1 and 2 are used in Chapter II which contains 73 theorems for a total of 194. It may be that the students had something to do with this large increase in the number of theorems.

On one occasion Chittenden was invited by Moore to give a summer course in topology at Texas. One of his interests was metrization but whether he chose this topic Moore did not say. In any case, after a few days Chittenden complained to Moore that they (the students) didn't know anything. Moore suggested that they be given some theorems to prove. After a couple of weeks, Chittenden complained that they had proved all the theorems he had planned for the summer. In this class were G.T. Whyburn, W.M. Whyburn, W.T. Reid, possibly Roberts and others. (One of the others was Lucille Smith, later Mrs. G.T. Whyburn.) With these students you just might run out of theorems unexpectedly.

There are spaces that satisfy the 1932 Axioms 1 and 2 (e.g. the Automobile Road Space) which are not metric but in which one can prove arcs and closed curves exist. In the second edition of his book (1962) there is only a trivial change in the 1932 Axiom 1; Ax.2 is the same; Ax.3 is unchanged except to unequivocally declare that every region contains more than one point; Ax.4 is unchanged (the Jordan Curve Theorem). In the 1916 paper Ax.4 said that space was locally compact. Moore wanted not to postulate the existence of compactness; so he threw out "local compactness", or did he? In his 1932 book Ax.5 says: If a is a point of a region R and b is a point distinct from a, there exists a simple closed curve in R separating a from b. Simple closed curves are compact. There follows the longest chapter in the book, 149 pages and 123 theorems. But then in Chapter V Axiom C says "space is locally compact". And this is unchanged in 1962. While Ax.5 is changed in 1962, it replaces the simple closed curve with a compact continuum. But in Chapter VI we get Axioms 51, and 52 which postulate no compactness. The reader must recall that Moore calls a set "a continuum" if it is closed and connected. It need not be compact.

Axiom 51. If p is a point of a region R, there exists a connected domain D containing p and bounded by a continuum T such that D + T is a subset of R.

Axiom 52. If D is a connected domain whose boundary is a continuum and ab is an arc lying wholly in D + b, then (D + b) - ab is connected.

Theorem 2 of Chapter VI of the 1962 book says: If p is a point of a region R, there exists a connected domain D containing p and bounded by a simple closed curve J such that J + D is a subset of R. So if we interpret "region" to mean simple region, one has the space satisfying the 1962 Axioms 1-5 on the basis of Axioms 1-51 and 52. These axioms assume no compactness nor do they assume the existence of arcs or simple closed curves. And Axioms 1 to 51 and 52 plus Ax.6: No compact continuum separates two noncompact point sets from each other, and Ax.7: "Space has a countable basis" (Moore says: "Space is completely separable") imply that space is topologically the plane or the 2-sphere depending upon whether Ax.8: "Space is not compact" or Axiom 8': "Space is compact" is true.

I have remarked elsewhere that among other things Moore in writing each of his books was also influenced by pedagogical motives (e.g., How well would it teach?) One often wonders how many were taught out of his 1916 paper, the 1932 book and the 1962 book? We have data on the number of students who took the Ph.D., in each time interval.

1916 paper:

3

at Penn (includes Kline and Mullikin)

 

6

at Texas (includes Wilder, Lubben, Whyburn (G.T) and Roberts)

1932 book:

24

at Texas (including Jones, Young, Bing, Anderson, Moise, Rudin, Burgess, Ball, Dyer, Hamstrom, and Armentrout)

1962 book:

17

at Texas (including Worrel, Howard Cook)

Total students:

50

 

As a guess probably 2/3 of those taking Moore's beginning topology course went on to the Ph.D. So perhaps there were approximately 75 students who took topology out of these three publications, These students didn't read the book. Moore read the book to them, leaving out the proofs and sometimes the examples, and they filled in the blanks. Many of these students later had research and teaching careers that were in some measure just as fruitful as Moore's, So it is no wonder that Alexandroff's "Topologie" (1935) in reviewing the work on continua and locally connected continua should remark: "In the hands of the Polish [School] (Mazurkiewicz, SierpiÕski, Kuratowski, Zarankiewicz, and others) as also the American School of R.L. Moore (Moore, Kline, Ayers, Gehman, Whyburn, Wilder and others) a very broad and extensive theory has developed."

1. C.E. Aull and R. Lowen eds., Handbook of the History of General Topology, Volume 1. 97-103. 1997 Kluwer Academic Publishers.


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