Moore was still vigorous and he disdained the word retirement. He continued to teach his full load of five organized courses but by law was allowed only half-time pay. He would not break the habit of teaching a complete sequence of courses each year, beginning with the calculus;" 10 and he continued teaching until It is notable that early in his career at Texas, he was appointed The University Research Lecturer for the year He was elected to the National Academy of Sciences in He was married to Margaret MacLellan Key in She survives him.
The work in geometry was mainly devoted to the axiomatic foundations.
Bibliography Items 1 and 2 were originally presented to the Society April 22, under the same title, Sets of metrical hypotheses for geometry. Item 1 shows that while Dehn proved that Hilbert's original axiom sets I, II and IV, augmented by the assertion, S, that the sum of the angles of a triangle is two right angles, are not sufficient to yield III parallels , nevertheless any space satisfying them i.
Item 2 was Moore's dissertation, and gives axioms for Euclidean geometry based on the primitive notions of point, order, and congruence, together with various alternatives. Item 3 is directed specifically at modification of Veblen's axioms. Item 5 is important for at least two reasons: I It shows that the space V satisfying axioms I-VIII of Veblen's thesis, and for which Veblen gave his proof of the Jordan Curve Theorem published 10 years before 12 is actually metrizable, in that V is topologically equivalent to the euclidean coordinate plane.
Item 17 is a thorough-going point set analysis of the euclidean and Bolyai-Lobachewskian planes in terms of point, region, and motion. It was inspired by Hilbert's Uber die Gundlagen der Geometrie, 13 and both the first six axioms and the methods are reminiscent of Item It differs from Hilbert's treatment in that it simultaneously analysis of the euclidean and Bolyai-Lobachewskian planes in terms of whereas Hilbert's analysis is largely confined to the group. The review Item 18 of the second volume of the Veblen-Young 14 work on projective geometry is mostly devoted to a critical analysis of its foundations, particularly as to whether a certain one of the defined terms should not really be an undefined term.
This seems to be the only review written by Moore; possibly the amount of time which he must have spent on it discouraged him from undertaking further reviews.
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One in whom the faculty for precision and logical structure is so well developed can usually find flaws in any book which will afford a challenge to make improvements. In this category we put Items 4, 6, 7, 16, 30, 34 and This is perhaps rather arbitrary, since all Moore's papers in point set theory bear relations, either direct or indirect, to analysis.
Item 16 gives necessary and sufficient conditions for a certain type of Frechet space to be compact in the covering sense , and Item 25 disproves a proposition stated by E. Hobson in his classic Theory of functions of a real variable.
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Item 30 is concerned with the relatively uniform convergence introduced by E. Moore, specifically with regard to functions defined on a measurable set. In deference to Moore's own preferences, we use this term even though many would use the term set-theoretic topology. Although, as noted above, signs of Moore's proceeding in this direction are to be found in his earlier work, they erupted full-blown in Item 10 On the foundations of plane analysis situs ; Item 9 is a preliminary announcement of Item Presented therein are three systems of axioms 1 , 2 , 3 , all based on point and region as primitive terms.
An outstanding feature of 1 and 2 is an axiom Axiom I which, in later terminology, implies regularity and existence of a countable base. This was well in advance of the work of Urysohn and others on metrization theorems in terms of these properties.
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System 3 does not imply metrizability, Axiom 1 being replaced by an axiom assuming only a countable base at each point. For a number of years both Moore and his students who had taken positions at other universities, used the system 1 as a basis for teaching their beginning courses in topology by the "Moore Method" or a suitable approximation thereto. The chief stumbling block in this course for students was Theorem 15 to the effect that every connected open set is arcwise connected.
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Indeed, one might say that the ability to prove this theorem was sort of an "open sesame" to Ph. And it was well known among the graduates of the "Moore School" that if one wanted to get a student into one of the universities at which Moore students were teaching, all that was needed was to be able to say of him, "He proved Theorem In Item 14, Moore showed that every space that satisfied either 1 or 2 is topologically equivalent to a number plane. Moore was to return later to the axiomatic characterization of the plane 2-sphere and its topology in both his book Item 51 and Item 53; in the latter the undefined terms were place which may be interpreted as bounded, connected, open set and a relation imbedded in.
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It was possibly inspired by the well-known Hahn-Mazurkiewicz topological characterization of the continuous curve defined analytically in Jordan's Cours d'analyse and proves the arcwise connectivity of such configurations. The report on continuous curves which Moore gave at a symposium in Lawrence, Kansas in , Item 27, covers the results which had been obtained up to this time so far as the topological nature of these curves was concerned. It is important not simply as a summary of results, but as indicating lines of research which were to be followed up by his own school especially G.
Whyburn 18 and others. Except for such items as 26, 44 and 45, however, and occasional theorems in other papers, Moore left the further investigation of continuous curves to his students, while concentrating himself on problems concerning general point sets, especially continua. In discussing the rest of Moore's publications in point set theory, it will be convenient to divide them into two classes, viz. By "structural" we indicate internal properties, and "positional" we indicate relations with an imbedding space. Items 28 and 31 extend a theorem of Sierpinski Item 31 turned out to be a "multiple" with Mazurkiewicz; cf.
Gross and Frechet. Item 41 was a contribution to the theory of Indecomposable continua, a type of configuration which received much attention from topologists in both the United States and Poland. The title of Item 43 is self-explanatory. Items are of particular interest in that they display a system of axioms whose primitive terms, in addition to point and region, contain the term contiguous to, denoting a relation between points.
In particular one point can be contiguous to another. Presumably a major reason for introducing this notion was for its application to structural properties of a continuum in terms of specialized subsets: for example, if the cyclic elements of a continuous curve C are regarded as "points" and two such points p and q are called contiguous if and only if one of the pair p, q is a point in the ordinary sense of the other, then C becomes an acyclic continuous curve in terms of its "points.
Items 57, 59, and 64 continue Moore's researches into the structure of continua, making special use of such concepts as continua of condensation and upper semicontinuous collections of continua. Upper semicontinuous collections were introduced in Item 38, where it was shown that if such a collection, G, of mutually exclusive bounded continua fills up a plane E and none of its elements separates E, then it is itself a plane in terms of the elements of G as "points" and with "limit point" suitably defined.
In view of the prescribed definition of limit for the elements of an upper semicontinuous collection, these elements are the counterimages of points of C under a monotonic continuous mapping of S onto C. This theorem was not only generalized to 2-manifolds and higher dimensional configurations, 20 but the notion of monotone mapping proved very fruitful in later set-theoretic investigations.
The notion of triod was introduced in Items 46 and One of the most striking results was the impossibility of imbedding an uncountable number of disjoint triods in the plane. Prime part decompositions, which had been introduced by H. Hahn, were exploited in Items 29 and The prime part notion was further extended and generalized by G.
Whyburn and the present writer. Item 12, A characterization of Jordan regions by properties having no reference to their boundaries, , was evidently principally inspired by 1 A. Schoenflies' classic work Die Entwickelung der Lehre von den Punktmannigfaltigkeiren Zweiter Tell, Leipzig, Teubner, in which, among other results concerning positional properties of plane continuous curves, conditions were given under which the common boundary of two plane domains will be a simple closed curve; and 2 Caratheodory's work on prime ends.
Like Caratheodory's condition, given for a like purpose, Moore's condition of "uniform connectedness im kleinen" applied to one domain alone; otherwise it is much simpler than the Caratheodory condition, and in the higher dimensional properties "ulc 11 " and "ULC 11 " has led to extensive generalizations.
Item 21 gives examples in three-dimensional space for which neither Moore's theorem nor the theorem of Schoenflies holds. In earlier work of Zoretti, F. Riesz, Schoenflies and Denjoy, it developed that every closed, bounded, totally disconnected plane point set is a subset of an are.
In Item 15, written jointly with J. Kline, 22 necessary and sufficient conditions were given in order that a plane point set should be a subset of an arc. This was later extended to n -dimensional space by E. Miller On subsets of a continuous curve which lie on an arc of the continuous curve.
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The concept of equicontinuous systems of curves was introduced in Item 20, and in Item 24 was used to characterize both closed 2-cells and open surfaces in three-dimensional space. And with reference to bounded plane domains that are complementary to continuous curves, Moore proved in Item 23 that their outer boundaries are simple closed curves; from this he was able to show that if two points are separated by a continuous curve, C, then they are separated by a simple closed curve of C.
Spirals were introduced in the plane in Item 68 and certain results established concerning sets of points on which a spiral may close down; e. Several of Moore's later doctoral students found further results concerning this notion.
Moore : Report on continuous curves from the viewpoint of analysis situs
Others of his positional papers, as their titles indicate, treat plane separation and accessibility. Item 44 establishes an interesting theorem to the effect that any two points in the complement of a plane continuous curve M can be joined by an arc that does not separate M. Moore's book, Foundations of point set theory, published in and, in revised form, in by the American Mathematical Society as Volume 13 in the Colloquium Publications Series, is based on the colloquium lectures which he gave before the Society in Boulder, Colorado, in August, The entire treatment, as might be expected, is based on an axiom system whose primitive terms are point and region.
In a way reminiscent of his fundamental paper Item 10 , it does not, however, use his original Axiom 1, but presents a new axiom presumably designed to accomplish most of the purposes of the original, but without implying that the space is metrizable or separable. On the basis of this axiom plus an axiom which states that every region is a point set , he proves theorems in the first chapter alone.
Additional axioms are added in subsequent chapters, but not until the last few pages of the book are axioms added which will ensure that the space is a plane or 2-sphere. Consequently the book is a very general compendium of point set results which has served, among other functions, as a reference point for many subsequent investigations. Although Moore's teaching and research have been treated separately above, in practice there was no true separation.
His own research results formed the basis of his teaching, and if they had no other justification, their use in this wise would have been sufficient. It is cause for conjecture just how much of his success as a teacher was due to his bringing his own ideas into class and allowing his students to participate in a reenactment of their creation; surely this experience must have created an empathy which the use of other materials, even using the "Method," would not have engendered.
It is pleasant to relate that The University of Texas has recognized his contributions by the dedication, in , of a new 17 level, double-winged building housing the departments of Astronomy, Mathematics and Physics, now known as Robert Lee Moore Hall.