Celebratio Mathematica

R H Bing

R H Bing: 1914–1986

by Michael Starbird

R H Bing loved to work on prob­lems in to­po­logy, per­haps be­cause he was con­sum­mately suc­cess­ful at solv­ing them. He was a math­em­atician of in­ter­na­tion­al renown, hav­ing writ­ten sem­in­al re­search pa­pers in gen­er­al and geo­met­ric to­po­logy. The Bing–Nagata–Smirnov met­riz­a­tion the­or­em is a fun­da­ment­al res­ult in gen­er­al to­po­logy that provides a char­ac­ter­iz­a­tion of which to­po­lo­gic­al spaces are gen­er­ated by a met­ric. His work and the meth­ods he used in the study of the geo­met­ric to­po­logy of 3-di­men­sion­al space were so sem­in­al and dis­tinct­ive that that area of in­vest­ig­a­tion is of­ten re­ferred to as Bing-type to­po­logy. His lead­er­ship both in re­search and in teach­ing res­ul­ted in his serving as pres­id­ent both of the Amer­ic­an Math­em­at­ic­al So­ci­ety and the Math­em­at­ic­al As­so­ci­ation of Amer­ica. However, we who knew him per­son­ally re­mem­ber him most for his zest for life that in­fec­ted every­one around him with a con­ta­gious en­thu­si­asm and good hu­mor. So this pa­per cel­eb­rates a life well lived, a life whose joy came partly from sig­ni­fic­ant con­tri­bu­tions to to­po­logy and partly from an over­flow­ing joie de vivre.

A vignette on Bing’s devotion to mathematics

It was a dark and stormy night, so R H Bing vo­lun­teered to drive some stran­ded math­em­aticians from the fogged-in Madis­on air­port to Chica­go. Freez­ing rain pel­ted the wind­screen and iced the road­way as Bing drove on — con­cen­trat­ing deeply on the math­em­at­ic­al the­or­em he was ex­plain­ing. Soon the wind­shield was fogged from the en­er­get­ic ex­plan­a­tion. The pas­sen­gers too had beaded brows, but their sweat arose from fear. As the math­em­at­ic­al de­scrip­tion got bright­er, the vis­ib­il­ity got dim­mer. Fi­nally, the con­fer­ees felt a trace of hope for their sur­viv­al when Bing reached for­ward — ap­par­ently to wipe off the mois­ture from the wind­shield. Their hope turned to hor­ror when, in­stead, Bing drew a fig­ure with his fin­ger on the foggy pane and con­tin­ued his proof — em­bel­lish­ing the il­lus­tra­tion with ar­rows and help­ful la­bels as needed for the demon­stra­tion.

Two of Bing’s math­em­at­ic­al col­leagues, Steve Ar­men­trout and C. E. Bur­gess, in­de­pend­ently re­called ver­sions of this mem­or­able even­ing. Those of us who knew Bing well avoided rais­ing math­em­at­ic­al ques­tions when he was driv­ing.

R H Bing star­ted and ended in Texas. He was born on Oc­to­ber 20, 1914, in Oak­wood, Texas, and there he learned the best of the dis­tinct­ively Texas out­look and val­ues. What he learned in Oak­wood guided him clearly throughout his life. He had a strong Texas drawl, which be­came more pro­nounced pro­por­tion­ate to his dis­tance from Texas; and he spoke a little louder than was ab­so­lutely ne­ces­sary for hear­ing alone. He might be called bois­ter­ous with the youth­ful vig­or and play­ful curi­os­ity that he ex­uded throughout his life. He was out­go­ing and friendly and con­tinu­ally found ways to make what he did fun. You could hear him from down the hall laugh­ing with his TAs while grad­ing cal­cu­lus ex­ams or do­ing oth­er work that deadens most people. He did not sleep well and when he woke at 4:00 a.m., he would get up and work. He es­pe­cially en­joyed work­ing on things re­quir­ing loud ham­mer­ing at that hour, on the grounds that if you are go­ing to be up at 4, the fam­ily should know about it. He prac­ticed the tra­di­tion­al Texas value of ex­er­cising in­de­pend­ent judg­ment, both in gen­er­al mat­ters and in mat­ters math­em­at­ic­al. He treated people kindly and gently — un­less he knew them, in which case it was more apt to be kindly and bois­ter­ously.

Both of Bing’s par­ents were in­volved in edu­ca­tion. His moth­er was a primary teach­er, and his fath­er was the su­per­in­tend­ent of the Oak­wood School Dis­trict. Bing’s fath­er died when R H was five, so Bing most re­membered his moth­er’s im­pact on his char­ac­ter and in­terests. Bing at­trib­uted his love for math­em­at­ics to his moth­er’s in­flu­ence. He re­called that she taught him to do men­tal arith­met­ic quickly and ac­cur­ately and to en­joy com­pet­i­tion both phys­ic­al and men­tal.

After high school Bing en­rolled in South­w­est Texas State Teach­ers Col­lege in San Mar­cos (now South­w­est Texas State Uni­versity) and re­ceived his B.A. de­gree in 1935 after two-and-a-half years there. Later in life Bing was named as the second dis­tin­guished alum­nus of South­w­est Texas State Uni­versity. The first per­son so honored was Lyn­don Baines John­son. Bing’s col­lege edu­ca­tion had pre­pared him as a high-school math­em­at­ics teach­er. He also was a high-jump­er on the track team, and could jump his own height — which was over 6 feet.

Bing’s fi­nal aca­dem­ic po­s­i­tion was as the Mil­dred Cald­well and Blaine Per­kins Kerr Centen­ni­al Pro­fess­or in Math­em­at­ics at the Uni­versity of Texas at Aus­tin, but his first aca­dem­ic ap­point­ment was as teach­er at Palestine High School in Palestine, Texas. There his du­ties in­cluded coach­ing the foot­ball and track teams, teach­ing math­em­at­ics classes, and teach­ing a vari­ety of oth­er classes, one of which was typ­ing. His meth­od of touch-typ­ing in­volved an­chor­ing his po­s­i­tion over the keys by keep­ing some con­stant pres­sure on his little fin­gers. This habit was hard to break, ap­par­ently, be­cause later he said that when he used an elec­tric type­writer or com­puter key­board (neither of which he did of­ten) he ten­ded to pro­duce large num­bers of ex­traneous “a’s.”

Nowadays one fre­quently hears com­plaints about a school sys­tem that gives the foot­ball coach the ad­ded as­sign­ment of teach­ing a math­em­at­ics class. One won­ders if those foot­ball boost­ers of a by­gone day in Palestine com­plained to the loc­al school board about a real math­em­at­ics teach­er coach­ing the foot­ball team.

In an ef­fort to im­prove pub­lic school edu­ca­tion in the 1930s, the Texas le­gis­lature ap­proved a policy whereby a teach­er with a mas­ter’s de­gree would re­ceive more pay than a teach­er with a bach­el­or’s de­gree. So, many teach­ers saved and scrimped dur­ing the nine-month ses­sion and went to sum­mer school dur­ing the three sum­mer months in an ef­fort to up­grade their tal­ents and their salar­ies. Bing was among them.

R H began pub­lic school teach­ing in 1935 and began tak­ing sum­mer school courses at the Uni­versity of Texas at Aus­tin. There he met Mary Blanche Hobbs, whom he mar­ried in 1938. R H and Mary en­joyed a long and happy mar­riage. They had four chil­dren: Robert Hobbs Bing, 1939; Susan Eliza­beth Bing, 1948; Vir­gin­ia Gay Bing, 1949; and Mary Pat Bing, 1952. His wife, Mary, all their chil­dren, their chil­dren’s spouses, and their grand­chil­dren still have fond memor­ies of Bing.

The same year as his mar­riage he earned a mas­ter of edu­ca­tion de­gree from UT. Dur­ing one sum­mer there, Bing took a course un­der the late Pro­fess­or R. L. Moore, also a mem­ber of the Na­tion­al Academy of Sci­ences. Moore was in­clined to de­prec­ate the ef­forts of an older stu­dent such as Bing was, so Bing had to prove him­self. But he was equal to the task.

Bing con­tin­ued to take some sum­mer courses while teach­ing in the high schools. In 1942 Moore was able to get Bing a teach­ing po­s­i­tion at the uni­versity, which al­lowed him to con­tin­ue gradu­ate study to work to­ward a doc­tor­ate and to try his hand at re­search.

An un­of­fi­cial rat­ing scheme some­times used by R. L. Moore and his col­leagues went something like this: You could ex­pect a stu­dent with Brown’s tal­ents and abil­it­ies every year; you could ex­pect a stu­dent with Lewis’s tal­ents and abil­it­ies once every 4 years; but a stu­dent with Smith’s tal­ents and abil­it­ies came along only once in 12 years. Bing’s tal­ents and abil­it­ies threw him in the 12-year class, or in an even high­er class, since he was one of the most dis­tin­guished math­em­aticians ever to have re­ceived his de­gree from the Uni­versity of Texas at Aus­tin. Sev­er­al of Moore’s later gradu­ate stu­dents have writ­ten that in the days after Bing, Moore used to judge his stu­dents by com­par­ing them with Bing — not to their ad­vant­age.

Bing re­ceived his Ph.D. in 1945, writ­ing his dis­ser­ta­tion on planar webs. Planar webs are to­po­lo­gic­al ob­jects now re­leg­ated to the ar­cana of his­tor­ic­al to­po­lo­gic­al ob­scur­ity. The res­ults from his dis­ser­ta­tion ap­peared in one of his earli­est pa­pers [1] (1946) in the Trans­ac­tions of the Amer­ic­an Math­em­at­ic­al So­ci­ety. He told us that the Trans­ac­tions had sent him 50 re­prints at the time and if we were in­ter­ested we could have some be­cause he still had 49 or so left.

But Bing did not have long to wait for re­cog­ni­tion of his math­em­at­ic­al tal­ent. He re­ceived his Ph.D. de­gree in May 1945, and in June 1945 he proved a fam­ous, long­stand­ing un­solved prob­lem of the day known as the Kline sphere-char­ac­ter­iz­a­tion prob­lem [2] (1946). This con­jec­ture states that a met­ric con­tinuum in which every simple closed curve sep­ar­ates but for which no pair of points sep­ar­ates the space is homeo­morph­ic to the 2-sphere.

When word spread that an un­known young math­em­atician had settled this old con­jec­ture, some people were skep­tic­al. Moore had not checked Bing’s proof, since it was his policy to cease to re­view the work of his stu­dents after they fin­ished their de­grees. Moore be­lieved that such re­view might tend to show a lack of con­fid­ence in their abil­ity to check the work them­selves. So when a fam­ous pro­fess­or wired Moore ask­ing wheth­er any first-class math­em­atician had checked the proof, Moore replied, “Yes, Bing had.”

Primar­ily be­cause of the renown among math­em­aticians gen­er­ated by his hav­ing solved a fam­ous con­jec­ture, Bing was offered po­s­i­tions at Prin­ceton Uni­versity and at the Uni­versity of Wis­con­sin, Madis­on. Moore nat­ur­ally wrote let­ters of re­com­mend­a­tion. One com­ment he made was that, al­though the Kline sphere-char­ac­ter­iz­a­tion prob­lem was a much bet­ter known top­ic than that of planar webs, Moore felt that it was Bing’s work on planar webs that demon­strated that Bing had the math­em­at­ic­al strength to be an out­stand­ing math­em­atician.

One of the lead­ing to­po­lo­gists of the time was at Prin­ceton, but Bing did not wish to fol­low in any­one’s foot­steps, so in 1947 he ac­cep­ted a po­s­i­tion at Wis­con­sin. He re­mained at Wis­con­sin for 26 years ex­cept for leaves: one at the Uni­versity of Vir­gin­ia (1949–50), three at the In­sti­tute for Ad­vanced Study in Prin­ceton (1957–58, 1962–63, 1967), one at the Uni­versity of Texas at Aus­tin (1971–72), and brief teach­ing ap­point­ments else­where. He re­turned to the Uni­versity of Texas at Aus­tin in 1973; but it was dur­ing his ten­ure at the Uni­versity of Wis­con­sin, Madis­on, that his most im­port­ant math­em­at­ic­al work was done, and his prom­in­ent po­s­i­tion in the math­em­at­ic­al com­munity es­tab­lished.

Bing’s early math­em­at­ic­al work primar­ily con­cerned top­ics in gen­er­al to­po­logy and con­tinua the­ory. He proved the­or­ems about con­tinua that are sur­pris­ing and still cent­ral to the field. Among these res­ults is Bing’s char­ac­ter­iz­a­tion of the pseudo-arc as a ho­mo­gen­eous in­decom­pos­able, chain­able con­tinuum [3] (1948). The res­ult that the pseudo-arc is ho­mo­gen­eous con­tra­dicted most people’s in­tu­ition about the pseudo-arc, and dir­ectly con­tra­dicted a pub­lished but er­ro­neous “proof” to the con­trary. Bing con­tin­ued to do some work in con­tinua the­ory throughout his ca­reer — in­clud­ing dir­ect­ing a Ph.D. dis­ser­ta­tion in the sub­ject at UT in 1977.

Around 1950 one of the great un­solved prob­lems in gen­er­al to­po­logy was the prob­lem of giv­ing a to­po­lo­gic­al char­ac­ter­iz­a­tion of the met­riz­ab­il­ity of spaces. In 1951 Bing gave such a char­ac­ter­iz­a­tion in his pa­per “Met­riz­a­tion of to­po­lo­gic­al spaces” in the Ca­na­dian Journ­al of Math­em­at­ics [4] (1951). Nagata and Smirnov proved sim­il­ar, in­de­pend­ent res­ults at about the same time, so now the res­ult is re­ferred to as the Bing–Nagata–Smirnov met­riz­a­tion the­or­em. That 1951 pa­per of Bing’s has prob­ably been re­ferred to in more pa­pers than any oth­er of his pa­pers, even though he later was iden­ti­fied with an al­to­geth­er dif­fer­ent branch of to­po­logy. Bing’s pa­per en­folds in a man­ner con­sist­ent with an im­port­ant strategy he prac­ticed in do­ing math­em­at­ics. He al­ways ex­plored the lim­its of any the­or­em he pro­posed to prove or un­der­stand. Con­sequently, he would ha­bitu­ally con­struct counter­examples to demon­strate the ne­ces­sity of each hy­po­thes­is of a the­or­em. In this pa­per Bing proved the­or­ems numbered 1 to 14 in­ter­spersed with ex­amples labeled A through H. The im­pact of this pa­per came both from his the­or­ems and from his counter­examples. Bing’s met­riz­a­tion the­or­ems de­scribe spaces with bases formed from count­able col­lec­tions of cov­er­ings, or spaces where open cov­ers have re­fine­ments con­sist­ing of count­able col­lec­tions of sets. He defined and dis­cussed screen­able spaces, strongly screen­able spaces, and per­fectly screen­able spaces — terms that have been largely re­placed by new ter­min­o­logy. He proved in this pa­per that reg­u­lar spaces are met­riz­able if and only if they are per­fectly screen­able. (The term per­fectly screen­able means that the space has a \( \omega_0 \)-dis­crete basis.)

His met­riz­a­tion the­or­ems hinged strongly on his un­der­stand­ing of a strong form of nor­mal­ity, and cer­tainly one of the legacies of this pa­per is his defin­i­tion of, and ini­tial ex­plor­a­tion of col­lec­tion­wise nor­mal­ity. This pa­per con­tains the the­or­em that a Moore space is met­riz­able if and only if it is col­lec­tion­wise nor­mal. After identi­fy­ing this im­port­ant prop­erty of col­lec­tion­wise nor­mal­ity, he ex­plored its lim­its by con­struct­ing an ex­ample of a nor­mal space that is not col­lec­tion­wise nor­mal. Bing was known for his ima­gin­at­ive nam­ing of spaces and con­cepts, but this ex­ample en­joys its en­dur­ing fame un­der the mundane monik­er of “Ex­ample G.” Im­me­di­ately fol­low­ing his de­scrip­tion of Ex­ample G, Bing in­cluded the fol­low­ing para­graph that formed the basis of count­less hours of fu­ture math­em­aticians’ labors:

One might won­der if Ex­ample G could be mod­i­fied so as to ob­tain a nor­mal de­velop­able space which is not met­riz­able. A de­velop­able space could be ob­tained by in­tro­du­cing more neigh­bor­hoods in­to the space [Ex­ample G]. However a dif­fi­culty might arise in in­tro­du­cing enough neigh­bor­hoods to make the res­ult­ing space de­velop­able but not enough to make it col­lec­tion­wise nor­mal.

Nowadays if you refer to Bing-type to­po­logy, you are re­fer­ring to a cer­tain style of geo­met­ric ana­lys­is of Eu­c­lidean 3-space that came to be as­so­ci­ated with Bing be­cause of the fun­da­ment­al work he did in the area and the dis­tinct­ive style with which he ap­proached it. The first pa­per Bing wrote in this area was titled “A homeo­morph­ism between the 3-sphere and the sum of two sol­id horned spheres” [5] and ap­peared in the An­nals of Math­em­at­ics. in 1952. It con­tains one of Bing’s best-known res­ults, namely that there are wild in­vol­u­tions of the 3-sphere, that is, it is pos­sible to re­flect 3-space through a mir­ror that is not to­po­lo­gic­ally em­bed­ded in the same man­ner as a flat plane. The res­ult in this pa­per hinges on a meth­od of shrink­ing geo­met­ric ob­jects in un­ex­pec­ted ways. When Bing first worked on the ques­tion con­sidered in this 1952 pa­per, he nat­ur­ally did not know wheth­er it was true or false. He claimed that he worked two hours try­ing to prove it was true, then two hours try­ing to prove it was false. When he ori­gin­ally worked on this prob­lem, he used col­lec­tions of rub­ber bands tangled to­geth­er in a cer­tain fash­ion to help him visu­al­ize the prob­lem. The math­em­at­ics that Bing did is ab­stract, but he claimed to get ideas about these ab­struse prob­lems from every­day ob­jects.

A fi­nal note about this prob­lem in­volves a pa­per that Bing wrote in 1984 con­tain­ing one of his last res­ults. If one shrinks the rub­ber bands in the man­ner de­scribed in Bing’s 1952 pa­per, each rub­ber band be­comes small in dia­met­er but very long. It be­came in­ter­est­ing to know wheth­er one could do a sim­il­ar shrink­ing without length­en­ing the bands — in oth­er words, could you do the same thing with string as Bing had proved could be done with rub­ber. Bing’s ori­gin­al pro­ced­ure had been stud­ied by nu­mer­ous gradu­ate stu­dents and re­search math­em­aticians for more than 30 years and yet no-one had been able to sig­ni­fic­antly im­prove Bing’s shrink­ing meth­od. It was left for Bing him­self to prove that “Shrink­ing without length­en­ing” (the title of this fi­nal pa­per) is pos­sible (1988) [25].

Bing’s res­ults in to­po­logy grew in num­ber and qual­ity. He proved sev­er­al land­mark the­or­ems, and then raised lots of re­lated ques­tions. Be­cause of his habit of rais­ing ques­tions, many oth­er math­em­aticians and stu­dents were able to prove good the­or­ems in the areas of math­em­at­ics that he pi­on­eered. He em­phas­ized the im­port­ance of rais­ing ques­tions in one’s pa­pers, and en­cour­aged his stu­dents and col­leagues to do so. He felt that math­em­aticians who read a pa­per are of­ten more in­ter­ested in what re­mains un­known than they are in­ter­ested in what has been proved.

The peri­od from 1950 un­til the mid-1960s was Bing’s most pro­duct­ive peri­od of re­search. He pub­lished about 115 pa­pers in his life­time, most dur­ing this peri­od at the Uni­versity of Wis­con­sin, Madis­on. In 1957 alone, three of his pa­pers ap­peared in the An­nals of Math­em­at­ics. These pa­pers con­cerned de­com­pos­i­tions of Eu­c­lidean 3-space [9], [10] and the the­or­em that sur­faces em­bed­ded in Eu­c­lidean 3-space can be ap­prox­im­ated by poly­hed­ral sur­faces [11]. Later, that res­ult was ex­ten­ded to show that the poly­hed­ral ap­prox­im­a­tion can be con­struc­ted to lie “mostly” on one side of the sur­face be­ing ap­prox­im­ated [20] (1963). In a 1958 An­nals pa­per [12], he proved that a com­pact 3-man­i­fold is homeo­morph­ic to \( \mathbb S^3 \) if and only if every simple closed curve is con­tained in a ball. This the­or­em was a par­tial res­ult in an at­tempt to settle the still-un­re­solved Poin­caré con­jec­ture in di­men­sion 3. In the next year, the An­nals pub­lished his in­de­pend­ent proof of the the­or­em that 3-man­i­folds can be tri­an­gu­lated [13] (1959), a res­ult that had re­cently been proved in a more com­plic­ated way by Ed­win Moise.

These the­or­ems and many oth­ers he proved about tame and wild sur­faces in Eu­c­lidean 3-space de­veloped the found­a­tions of the in­vest­ig­a­tion of the geo­met­ric to­po­logy of 3-space. Bing stated and proved ba­sic facts about 3-space and how sur­faces can lie in it. He proved that a sur­face is tame if it can be ap­prox­im­ated en­tirely from the side [15] (1959). He proved that every sur­face in 3-space con­tains tame arcs [19] (1962) and every sur­face in 3-space can be pierced by a tame arc [18] (1962).

Along the way Bing pro­duced many in­triguing ex­amples, many with mem­or­able nick­names: “The Bing sling” — a simple closed curve that pierces no disk [8] (1956); “Bing’s sticky-foot to­po­logy” — a con­nec­ted count­able Haus­dorff space [6] (1953); “Bing’s hooked rug” — a wild 2-sphere in 3-space that con­tains no wild arc [17] (1961). These ex­amples helped show the lim­its of what is true.

His re­search suc­cess brought him hon­ors, awards, and re­spons­ib­il­it­ies. He was quickly pro­moted through the ranks at the Uni­versity of Wis­con­sin, be­com­ing a Rudolph E. Langer Re­search Pro­fess­or there in 1964. He was a vis­it­ing lec­turer of the Math­em­at­ic­al As­so­ci­ation of Amer­ica (1952–53, 1961–62) and the Hedrick lec­turer for the Math­em­at­ic­al As­so­ci­ation of Amer­ica (1961). He was chair­man of the Wis­con­sin Math­em­at­ics De­part­ment from 1958 to 1960, but ad­min­is­trat­ive work was not his fa­vor­ite. He was pres­id­ent of the Math­em­at­ic­al As­so­ci­ation of Amer­ica (1963–64).

In 1965 he was elec­ted to mem­ber­ship in the Na­tion­al Academy of Sci­ences. He was chair­man of the Con­fer­ence Board of Math­em­at­ic­al Sci­ences (1966–67) and a U.S. del­eg­ate to the In­ter­na­tion­al Math­em­at­ic­al Uni­on (1966, 1978). He was on the Pres­id­ent’s Com­mit­tee on the Na­tion­al Medal of Sci­ence (1966–67, 1974–76), chair­man of the Di­vi­sion of Math­em­at­ics of the Na­tion­al Re­search Coun­cil (1967–69), mem­ber of the Na­tion­al Sci­ence Board (1968–75), chair­man of the Math­em­at­ics Sec­tion of the Na­tion­al Academy of Sci­ences (1970–73), on the Coun­cil of the Na­tion­al Academy of Sci­ences (1977–80), and on the Gov­ern­ing Board of the Na­tion­al Re­search Coun­cil (1977–80). He was a col­loqui­um lec­turer of the Amer­ic­an Math­em­at­ic­al So­ci­ety in 1970. In 1974 he re­ceived the Dis­tin­guished Ser­vice to Math­em­at­ics Award from the Math­em­at­ic­al As­so­ci­ation of Amer­ica. He was pres­id­ent of the Amer­ic­an Math­em­at­ic­al So­ci­ety in 1977–78. He re­tired from the Uni­versity of Texas at Aus­tin in 1985 as the Mil­dred Cald­well and Blaine Per­kins Kerr Centen­ni­al Pro­fess­or in Math­em­at­ics. He re­ceived many oth­er hon­ors and served in many oth­er re­spons­ible po­s­i­tions throughout his ca­reer. He lec­tured in more than 200 col­leges and uni­versit­ies in 49 states and in 17 for­eign coun­tries.

Bing be­lieved that math­em­at­ics should be fun. He was op­posed to the idea of for­cing stu­dents to en­dure math­em­at­ic­al lec­tures that they did not un­der­stand or en­joy. He liked to work math­em­at­ics out for him­self and thought that stu­dents should be giv­en the op­por­tun­ity to work prob­lems and prove the­or­ems for them­selves. Dur­ing his years in Wis­con­sin, Bing dir­ec­ted a very ef­fect­ive train­ing pro­gram for fu­ture to­po­lo­gists. The first-year gradu­ate to­po­logy class, which he of­ten taught there, would some­times num­ber 40 or more stu­dents. He dir­ec­ted the Ph.D. dis­ser­ta­tions of 35 stu­dents and in­flu­enced many oth­ers dur­ing par­ti­cip­a­tion in sem­inars and re­search dis­cus­sions.

Bing en­joyed teach­ing and felt that ex­per­i­ments in teach­ing were usu­ally suc­cess­ful — not be­cause the new meth­od was ne­ces­sar­ily bet­ter but be­cause do­ing an ex­per­i­ment showed an in­terest in the stu­dents, which they ap­pre­ci­ated and re­spon­ded to. Here are a couple of the ex­per­i­ments he tried while teach­ing at UT. Bing thought that a per­son who could solve a prob­lem quickly de­served more cred­it than a per­son who solved it slowly. He would say that an em­ploy­er would rather have an em­ploy­ee who could solve two prob­lems in as much time as it took for someone else to solve one. So in some of his un­der­gradu­ate classes he in­tro­duced speed points. For a 50-minute test he gave an ex­tra point for each minute the test was sub­mit­ted early. He no­ticed that of­ten the people who did the work the quick­est also were the most ac­cur­ate. Speed points were some­what pop­u­lar, and some­times he would let the class vote on wheth­er speed points would be used on a test. An­oth­er ex­per­i­ment in test-giv­ing was not pop­u­lar. One day Bing pre­pared a cal­cu­lus test that he real­ized was too long. In­stead of de­let­ing some ques­tions, however, he de­cided to go ahead and give the test, but as he phrased it, “Let every­one dance to the tune of their own drum­mer.” That is, each per­son could do as many or as few of the prob­lems as he or she wished and would be graded on the ac­cur­acy of the prob­lems sub­mit­ted. The class was quite angry when the highest score was ob­tained by a per­son who had at­temp­ted only one prob­lem.

In the 1971–72 school year, Bing ac­cep­ted an of­fer to vis­it the De­part­ment of Math­em­at­ics at the Uni­versity of Texas at Aus­tin. In 1973 the math­em­at­ics de­part­ment per­suaded Bing to ac­cept a per­man­ent po­s­i­tion at UT. Bing be­lieved that part of the fun of life was to take on a vari­ety of chal­lenges. When he ac­cep­ted the po­s­i­tion at UT, he came with the idea of build­ing the math­em­at­ics de­part­ment in­to one of the top 10 state-uni­versity math­em­at­ics de­part­ments in the coun­try. While he was at Texas, from 1973 un­til his death in 1986, he helped to im­prove the re­search stand­ing of the de­part­ment by re­cruit­ing new fac­ulty and by help­ing to change the at­ti­tudes and ori­ent­a­tion of the ex­ist­ing fac­ulty. Rais­ing re­search stand­ards was the watch­word of that peri­od, and is the guid­ing prin­ciple for the math­em­at­ics de­part­ment now. Bing was chair­man of the de­part­ment from 1975 to 1977, but he used his in­ter­na­tion­al prom­in­ence for re­cruit­ing pur­poses throughout his stay at UT. The De­part­ment of Math­em­at­ics was con­sidered one of the most-im­proved de­part­ments over the peri­od of Bing’s ten­ure. The 1983 re­port of the Con­fer­ence Board of As­so­ci­ated Re­search Coun­cils lis­ted Texas as the second most-im­proved math­em­at­ics de­part­ment in re­search stand­ing dur­ing the peri­od 1977–82, rank­ing it num­ber 14 among state-uni­versity math­em­at­ics de­part­ments at that time. The strategy of re­search im­prove­ment has con­tin­ued in the UT De­part­ment of Math­em­at­ics through the present day, and Bing would cer­tainly be proud to see the de­part­ment’s con­tin­ued im­prove­ment in its re­search stature.

Bing ac­com­plished much dur­ing his life, and left us with many ideas, per­son­al and math­em­at­ic­al, to con­sider and en­joy. He left to­po­lo­gists a treas­ure-trove of the­or­ems and tech­niques, and left the UT De­part­ment of Math­em­at­ics with a goal and 13 years of good pro­gress to­ward it. He was a man of strong char­ac­ter and in­teg­rity, who liked to un­der­stand things for him­self. For ex­ample, he nev­er claimed to un­der­stand a the­or­em un­less he per­son­ally knew a proof of it. He made de­cisions based on his own ex­per­i­ence, re­ly­ing on his in­de­pend­ent judg­ment of a per­son or a cause whenev­er pos­sible, rather than av­er­aging the opin­ions of oth­ers. He was a kind man, and re­spec­ted people for their own mer­its rather than meas­ur­ing them on a single scale.

R H Bing died on April 28, 1986. He suffered from can­cer and heart trouble dur­ing his last years, but he nev­er com­plained about his health prob­lems, nor did he al­low dis­com­fort to dampen his en­thu­si­asm and good spir­its. He was an ex­em­plary per­son. His friends, his fam­ily, his stu­dents, and the math­em­at­ic­al com­munity have been en­riched bey­ond bound by his char­ac­ter, his wis­dom, and his un­fail­ing good cheer, and con­tin­ue to be en­riched by his memory.


[1]R. H. Bing: “Con­cern­ing simple plane webs,” Trans. Am. Math. Soc. 60 : 1 (July 1946), pp. 133–​148. See also Bing’s PhD thes­is (1945). MR 0016646 Zbl 0060.​40310 article

[2]R. H. Bing: “The Kline sphere char­ac­ter­iz­a­tion prob­lem,” Bull. Am. Math. Soc. 52 : 8 (1946), pp. 644–​653. MR 0016645 Zbl 0060.​40501 article

[3]R. H. Bing: “A ho­mo­gen­eous in­decom­pos­able plane con­tinuum,” Duke Math. J. 15 : 3 (1948), pp. 729–​742. MR 0027144 Zbl 0035.​39103 article

[4]R. H. Bing: “Met­riz­a­tion of to­po­lo­gic­al spaces,” Ca­na­dian J. Math. 3 (1951), pp. 175–​186. MR 0043449 Zbl 0042.​41301 article

[5]R. H. Bing: “A homeo­morph­ism between the 3-sphere and the sum of two sol­id horned spheres,” Ann. Math. (2) 56 : 2 (September 1952), pp. 354–​362. MR 0049549 Zbl 0049.​40401 article

[6]R. H. Bing: “A con­nec­ted count­able Haus­dorff space,” Proc. Am. Math. Soc. 4 : 3 (1953), pp. 474. MR 0060806 Zbl 0051.​13902 article

[7]R. H. Bing: “Loc­ally tame sets are tame,” Ann. Math. (2) 59 : 1 (January 1954), pp. 145–​158. MR 0061377 Zbl 0055.​16802 article

[8]R. H. Bing: “A simple closed curve that pierces no disk,” J. Math. Pures Ap­pl. (9) 35 (1956), pp. 337–​343. MR 0081461 Zbl 0070.​40203 article

[9]R. H. Bing: “Up­per semi­con­tinu­ous de­com­pos­i­tions of \( E^3 \),” Ann. Math. (2) 65 : 2 (March 1957), pp. 363–​374. MR 0092960 Zbl 0078.​15201 article

[10]R. H. Bing: “A de­com­pos­i­tion of \( E^3 \) in­to points and tame arcs such that the de­com­pos­i­tion space is to­po­lo­gic­ally dif­fer­ent from \( E^3 \),” Ann. Math. (2) 65 : 3 (May 1957), pp. 484–​500. MR 0092961 Zbl 0079.​38806 article

[11]R. H. Bing: “Ap­prox­im­at­ing sur­faces with poly­hed­ral ones,” Ann. Math. (2) 65 : 3 (May 1957), pp. 465–​483. Ex­pan­ded ver­sion of an art­icle in Sum­mer in­sti­tute on set the­or­et­ic to­po­logy (1957). MR 0087090 Zbl 0079.​38805 article

[12]R. H. Bing: “Ne­ces­sary and suf­fi­cient con­di­tions that a 3-man­i­fold be \( S^3 \),” Ann. Math. (2) 68 : 1 (July 1958), pp. 17–​37. MR 0095471 Zbl 0081.​39202 article

[13]R. H. Bing: “An al­tern­at­ive proof that 3-man­i­folds can be tri­an­gu­lated,” Ann. Math. (2) 69 : 1 (January 1959), pp. 37–​65. MR 0100841 Zbl 0106.​16604 article

[14]R. H. Bing: “The Cartesian product of a cer­tain non­man­i­fold and a line is \( E^4 \),” Ann. Math. (2) 70 : 3 (November 1959), pp. 399–​412. Ex­pan­ded ver­sion of an art­icle in Bull Am. Math. Soc. 64:3 (1958). MR 0107228 Zbl 0089.​39501 article

[15]R. H. Bing: “Con­di­tions un­der which a sur­face in \( E^3 \) is tame,” Fund. Math. 47 : 1 (1959), pp. 105–​139. MR 0107229 Zbl 0088.​15402 article

[16]R. H. Bing: “A sur­face is tame if its com­ple­ment is 1-ULC,” Trans. Am. Math. Soc. 101 : 2 (November 1961), pp. 294–​305. MR 0131265 Zbl 0109.​15406 article

[17]R. H. Bing: “A wild sur­face each of whose arcs is tame,” Duke Math. J. 28 : 1 (1961), pp. 1–​15. MR 0123302 Zbl 0101.​16507 article

[18]R. H. Bing: “Each disk in \( E^3 \) is pierced by a tame arc,” Am. J. Math. 84 : 4 (October 1962), pp. 591–​599. MR 0146812 Zbl 0178.​27202 article

[19]R. H. Bing: “Each disk in \( E^3 \) con­tains a tame arc,” Am. J. Math. 84 : 4 (October 1962), pp. 583–​590. MR 0146811 Zbl 0178.​27201 article

[20]R. H. Bing: “Ap­prox­im­at­ing sur­faces from the side,” Ann. Math. (2) 77 : 1 (January 1963), pp. 145–​192. MR 0150744 Zbl 0115.​40603 article

[21]R. H. Bing: “In­equi­val­ent fam­il­ies of peri­od­ic homeo­morph­isms of \( E^3 \),” Ann. Math. (2) 80 : 1 (July 1964), pp. 78–​93. MR 0163308 Zbl 0123.​16801 article

[22]R. H. Bing and K. Bor­suk: “Some re­marks con­cern­ing to­po­lo­gic­ally ho­mo­gen­eous spaces,” Ann. Math. (2) 81 : 1 (January 1965), pp. 100–​111. MR 0172255 Zbl 0127.​13302 article

[23]R. D. An­der­son and R. H. Bing: “A com­plete ele­ment­ary proof that Hil­bert space is homeo­morph­ic to the count­able in­fin­ite product of lines,” Bull. Am. Math. Soc. 74 : 5 (1968), pp. 771–​792. MR 0230284 Zbl 0189.​12402 article

[24]R. H. Bing: The geo­met­ric to­po­logy of 3-man­i­folds. AMS Col­loqui­um Pub­lic­a­tions 40. Amer­ic­an Math­em­at­ic­al So­ci­ety (Provid­ence, RI), 1983. MR 728227 Zbl 0535.​57001 book

[25]R. H. Bing: “Shrink­ing without length­en­ing,” To­po­logy 27 : 4 (1988), pp. 487–​493. MR 976590 Zbl 0673.​57011 article