John Henderson Roberts, the 7th Ph.D. student of R. L. Moore, died at the Carolina Meadows Health Center in Chapel Hill, NC on October 8, 1997. He had been confined to a wheelchair for several years due to the ill effect of several strokes. He was 91 years old when he died. He is survived by his son, John Edward, grandson George, and great-granddaughters Monica (age 5) and Susan (age 2).

Roberts was born in Raywood, Texas (about 40 miles east of Houston) on September 2, 1906. He received the A.B. degree in mathematics from the University of Texas in 1927 at the age of 21. Two years later in 1929 he received the Ph.D. degree. Roberts related to us that he was able to earn the Ph.D. so quickly because he made such a favorable impression in Moore's introductory topology course, that Moore immediately moved him into the advanced class. Roberts said that this event was "indelibly etched in my mind." Moore's confidence was quickly justified: In his first four papers, Roberts answered questions raised by C. Kuratowski, G. T. Whyburn, K. Menger and R. L. Moore, and by 1933, just four years after receiving his degree, he had published 14 papers.

Gian-Carlo Rota, in his recent book Indiscrete Thoughts, states that mathematicians can be subdivided into two types: problem solvers and theorizers (page 45). Roberts falls squarely into the problem solver division. Indeed he was known to say "I am a sucker for a good problem."

Roberts visited the University of Pennsylvania during 1929-30, where he worked with J. R. Kline, Moore's first Ph.D. student. During 1930-31 he was adjunct professor at the University of Texas. Due to cutbacks caused by the Great Depression, Roberts was not able to stay at the University of Texas. In 1931 he moved to Duke University where he remained a member of the faculty at Duke until he retired in 1971. During World War II, he served as a lieutenant commander in the Navy.

Roberts met his future wife, Doretta von Boeckman (1904-1988), a native of Austin, Texas, while he was attending the University of Texas. They were married on August 27, 1928. Roberts was a boarder at the von Boeckmann's house while a student at Texas, and Roberts was projectionist at a theater in Austin where Doretta played the piano for silent movies.

Professor and Mrs. Roberts were always very kind to new faculty members and graduate students. They gave frequent Saturday night parties which were greatly appreciated by all. Mary Ellen and Walter Rudin remember going to many parties at the Roberts' house. Walter had been going to the parties as a graduate student at Duke. He received the Ph.D degree from Duke in June, 1949, and was hired as an instructor. The following September Mary Ellen Estill was hired as an instructor at Duke having just received the Ph.D. degree from the University of Texas. They recall their first "dates" were probably to the Roberts' parties. These parties were unique at Duke, the only departmental social occasions where all of the graduate students and the faculty could meet. Mary Ellen said "John Roberts was very much a mentor for me at Duke and for Walter and other graduate students, and Doretta cheered us all on too. They were also enthusiastic over a possible budding romance between Walter and me but certainly not pushy about it."

At the parties we attended in the early 1960's, Roberts showed us numerous electronic devices that he built from kits. This included amateur radio equipment, a color television set, and an appliance timer, which Roberts gave to Jerry who is still using it. To Mrs. Roberts' delight, Professor Roberts added a switch on a long cord that could, from across the room, turn the sound from the television off and on while leaving the picture visible. She called this device her "blab-off."

Roberts will be remembered by his colleagues as a remarkable and distinguished mathematician, and in addition by his students as an unselfish and friendly thesis advisor.

The phrase "he had a long and distinguished career," certainly applies to Roberts. During his 40 years at Duke, he had 24 Ph.D. students, was director of graduate studies for the department from 1948-1960, and was chair of the mathematics department from 1966-68. He served as managing editor of the Duke Mathematical Journal from 1951-1960, and was Secretary of the American Mathematical Society in 1954. He spent the academic year 1937-38 at Princeton and wrote a joint paper with N. Steenrod while there.

Early in his career Roberts worked mainly on the topic of connectivity (especially in the plane), but beginning in the 1940's, his interests shifted, and he began to work in dimension theory. In the early 1950's, however, he wrote two papers on integral equations which resulted from questions posed to him. During the last six years of his career, he worked on the area of metric-dependent dimension functions with K. Nagami, from Ehime University in Japan. Nagami visited Duke University during the academic years 1963-65. Nagami and Roberts made a very good team. Nagami, an expert in dimension theory (and the oriental board game Go), had the knack for finding numerous interesting open problems in dimension theory. Their joint work, which might be characterized as the collaboration between a theorizer and a problem solver, contributed greatly to the theory of metric-dependent dimension functions.

We will discuss briefly a small sample of Roberts' mathematics.

• In his Ph.D. dissertation (published in 1929) Roberts proved that if M is a plane continuum which contains no open set, then there exists a set G of simple closed curves filling the whole plane such that every curve g in G intersects M in either the empty set or a totally disconnected set (this improved a result of K. Menger).

• In 1948, Roberts answered a question raised in the famous book "Dimension Theory" by Hurewicz and Wallman. A well-known theorem in dimension theory says that if X is a separable metric space of dimension n and Y is a (2n+1)-dimensional Euclidean cube, then the set H of all homeomorphism from X into Y, considered as a subset of C(X,Y), the space of all continuous functions from X to Y with the sup norm metric, contains a dense G-delta set. Moreover, if X is compact, then H is a (dense) G-delta set. The question raised in the book was whether H is always a G-delta set in the non-compact case. Roberts gave a negative answer to this question.

• In 1940 P. Erdos proved that the set R of all points in Hilbert space with all coordinates rational has dimension 1. It follows immediately from the theorem in dimension theory mentioned above that R can be embedded into Euclidean 3-space. Erdos asked if R could be embedded into the plane. In 1956, Roberts answered this in the affirmative.

• In 1959 J. Stallings proved that there exists a compact 0-dimensional subset of the unit square that intersects the graph of every continuous function from the closed unit interval into itself, and he asked if the same kind of set could be found that intersects the graph of every connectivity function (a function is called a connectivity function if for every connected subset of its domain, the graph of the function over the set is a connected set). In 1965 Roberts solved this problem in the affirmative.

• In 1959-61, Hans Debrunner, a Swiss mathematician and an expert in combinatorial geometry visited Duke for the academic year, having spent the previous year at Princeton with R. H. Fox. One day he mentioned a geometry problem to Roberts which, according to Debrunner, had been unsolved for several years. When Roberts saw the problem he said "I can't believe that problem hasn't been solved." He then wrote up a solution and sent to Elemente de Mathematik, the journal in which the problem first appeared. Roberts proved that if a triangle DEF is inscribed in a triangle ABC with D on BC, E on CA and F on AB then the minimum of the perimeters of the four smaller triangles is always assumed by a corner triangle. The only case in which the triangle DEF assumes this minimum is when all four of the smaller triangles are congruent.

• In 1967, Nagami and Roberts introduced several dimension functions for metric spaces which depend on the particular metric given for the space (metric-dependent dimension functions). For example, the one called d_2 is defined as follows: d_2 assigns the empty set the value -1, and for a non-empty space X, d_2(X) is not more than n provided for any n+1 pairs of closed sets, each pair at a positive distance apart, there exists n+1 closed sets, one separating each pair of closed sets, and having no point in common to all n+1 of them. They proved various relations among their dimension functions, the classical covering dimension (dim), and metric dimension (\mu-dim). They constructed an example of a metric space R such that d_2(R) < dim(R) < \mu-dim(R). This example improved the result of K. Sitnikov that the metric dimension and covering dimension are distinct dimension functions.

• In his last published paper in 1970 (written with F. G. Slaughter, Jr.) Roberts combined the two main interests in his career by giving a characterization of covering dimension being greater than or equal to n, in terms of the existence of a certain continuum.

1940: Paul Wilner Gilbert, and Abram Venable Martin , Jr.
1942: Paul Civin
1948: Samuel Wilfred Han
1949: Ivey Clenton Gentry, and Milton Preston Jarnagin, Jr.
1950: Lewis McLeod Fulton, Jr.
1952: Henry Sharp, Jr
1955: William R. Smythe
1958: Arthur L. Gropen, and Auguste Forge
1959: Nosup Kwak
1960: M. Jawad Saadaldin
1962: Richard E. Hodel
1963: Richardson King, and George M. L. Rosenstein
1964: Bruce Richard Wenner
1965: Jerry E. Vaughan
1966: Frank Gill Slaughter, Jr., and James Wilkinson
1967: James C. Smith,
1968: Leonard E. Soniat
1970: Glenn A. Bookhout, and Joseph C. Nichols

1. On a problem of C. Kuratowski concerning upper semi- continuous collections, Fundamenta Mathematicae 14(1929), 96-102.
2. (with J. L. Dorroh) On a problem of G. T. Whyburn, Fundamenta Mathematicae 13(1929), 58-61.
3. On a problem of Menger concerning regular curves, Fundamenta Mathematicae 14(1929), 327-333.
4. Concerning atroidic continua, Monatsheften fur Mathematik und Physik 37(1930), 223-230.
5. A note concerning cactoids, Bulletin of the American Mathematical Society 36(1930), 894-896.
6. Concerning collections of continua not all bounded, American Journal of Mathematics 52(1930), 551-562.
7. Concerning non-dense plane continua, Transactions of the American Mathematical Society 32(1930), 6-30.
8. A non-dense plane continuum, Bulletin of the American Mathematical Society 37(1931), 720-722.
9. A point set characterization of closed two-dimensional manifolds, Fundamenta Mathematicae 18(1931), 39-46.
10. Concerning metric collections of continua, American Journal of Mathematics 53(1931), 422-426.
11. Concerning topological transformations in E^n, Transactions of the American Mathematical Society 34(1932), 252-262.
12. Concerning uniordered spaces, Proceedings of the National Academy of Sciences 18(1932), 403-406.
13. A property related to completeness, Bulletin of the American Mathematical Society 38(1932), 835-838.
14. Concerning compact continua in certain spaces of R. L. Moore, Bulletin of the American Mathematical Society 39(1933), 615-621.
15. On a problem of Knaster and Zarankiewicz, Bulletin of the American Mathematical Society 40(1934), 281-283.
16. Collections filling a plane, Duke Mathematical Journal 2(1936), 10-19.
17. (with N. E. Steenrod) Monotone transformations of 2- dimensional manifolds, Annals of Mathematics 39(1938), 851-862.
18. Note on topological mappings, Duke Mathematical Journal 5(1939), 428-430.
19. Two-to-one transformations, Duke Mathematical Journal 6(1940), 256-262.
20. A theorem on dimension, Duke Mathematical Journal 8(1941), 565- 574.
21. (with A. V. Martin) Two-to-one transformations on two-manifolds, Transactions of the American Mathematical Society 49(1941), 1-17.
22. (with Paul Civin) Sections of continuous collections, Bulletin of the American Mathematical Society 49(1943), 142-143.
23. Open transformations and dimension, Bulletin of the American Mathematical Society 53(1947), 176-178.
24. A problem in dimension theory, American Journal of Mathematics 70 (1948), 126-128.
25. (with W. R. Mann) On a certain nonlinear integral equation of the Volterra type, Pacific Journal of Mathematics 1(1951), 431-445.
26. A nonconvergent iterative process, Proceedings of the American Mathematical Society 4(1953), 640-644.
27. The rational points in Hilbert space, Duke Mathematical Journal 23(1956), 489-492.
28. Problem of Treybig concerning separable spaces, Duke Mathematical Journal 28(1961), 153-156.
29. Contractibility in spaces of homeomorphisms, Duke Mathematical Journal 28 (1961), 213-220.
30. Solution to Aufgabe 260(second part), Elemente Der Mathematik 16(1961), 109-111.
31. (with L. R. King and G. M. Rosenstein, Jr.) Concerning some problems raised by Lelek, Fundamenta Mathematicae 54(1964), 325-334.
32. (with Keio Nagami) A note on countable-dimensional metric spaces, Proceedings of the Japan Academy 41(1965), 155-158.
33. Zero-dimensional sets blocking connectivity functions, Fundamenta Mathematicae 57(1965), 173-179.
34. (with Keio Nagami) Metric-dependent dimension functions, Proceedings of the American Mathematical Society 16(1965), 601-604.
35. (with Keio Nagami) A study of metric-dependent dimension functions, Transactions of the American Mathematical Society, 129 (1967), 414-435.
36. (with F. G. Slaughter, Jr.) Metric dimension and equivalent metrics, Fundamenta Mathematicae, 62 (1968), 1-5.
37. Realizability of metric-dependent dimension, Proceedings of the American Mathematical Society, 19 (1968), 1439-1442.
38. Metric-dependent function d_2, and covering dimension, Duke Mathematical Journal, 38 (1970), 467-472.
39. (with F. G. Slaughter, Jr.), Characterization of dimension in terms of the existence of a continuum, Duke Mathematical Journal, 37 (1970), 681- 688.