Celebratio Mathematica

“Many years ago, I was for­tu­nate enough to study un­der R H Bing at the Uni­versity of Wis­con­sin. His ap­proach to teach­ing was most un­usu­al and I re­mem­ber one in­cid­ent in par­tic­u­lar.

“Early one morn­ing I was awakened by Pro­fess­or Bing and asked if I would like to join him trav­el­ing to a meet­ing about 200 miles away. I and a group of oth­er stu­dents sat in Pro­fess­or Bing’s car, while he drove at high speed down a two-lane road. Un­for­tu­nately, the win­dows were fogged up, but Pro­fess­or Bing took the op­por­tun­ity to use these win­dows as black­boards while driv­ing.

“At these time, the math No­tices had short re­search ab­stracts. So, dur­ing the trip we were quizzed as to the pos­sible de­tails of the res­ults. At one point I re­mem­ber blurt­ing out something and Pro­fess­or Bing was de­lighted that I might have filled in some de­tail of a proof. Ac­tu­ally, I was try­ing to say that we just sped through a stop sign.”

“Rob Kirby asked me to write a few sen­tences re­mem­ber­ing the great math­em­atician R H Bing. I’m de­lighted to do so. Bing did his work blessed with a cer­tain pur­ity of vis­ion and breath­tak­ing ori­gin­al­ity. What comes to my mind, when I think of him is no spe­cif­ic an­ec­dote, but rather a mood of thought: the vi­va­city of how he presen­ted his ideas, and the way he savored ideas.

“His geo­met­ric in­sight was so power­ful, so cap­able of be­ing dir­ectly con­veyed — al­most with no tech­nic­al dress­ing. It is the rare newly-found math­em­at­ic­al ex­plan­a­tion that can be told in a single phrase; but Bing was the source of many that could. I don’t know who came up with the mar­velous hint think of a light-bulb hung from the ceil­ing for the proof that it is im­possible to ‘knot’ an arc that goes from one bound­ary com­pon­ent to an­oth­er in \( \mathbb{S}^2\times [0, 1] \). Even if this was not Bing’s idea it ra­di­ates a Bing-like style—burst­ing, as it does, with such sim­pli­city and ur­gency.

“Per­haps ir­rel­ev­antly, I also can’t help think­ing of the mu­sic­al­ity of his ex­plan­a­tions. Now, surely, the way in which he pro­nounced the phrase ‘Ant­oine’s Neck­lace’ — with a linger­ing Texas twang to the ‘toine’ part — has noth­ing to do with his math­em­at­ics, but it works well with the way he he presen­ted his ideas — songs that vi­brate with sheer geo­met­ric in­sight. What muse of geo­metry offered those gifts to him? Each of his fam­ous res­ults ex­pands our no­tion of what to­po­logy can do. And, think­ing about them half a cen­tury after they were con­ceived, they re­main no less astound­ing: Bing’s the­or­ems are news that stays news. Who else would ima­gine that the to­po­lo­gic­al three-di­men­sion­al sphere could be rep­res­en­ted as the double of the bad-com­pon­ent com­ple­ment of the Al­ex­an­der Horned Sphere?”

“R. L. Moore was proud of his stu­dents and he was es­pe­cially proud of R H Bing.

“Shortly after he re­ceived his Ph.D., Bing solved a long-out­stand­ing and dif­fi­cult con­jec­ture known as the Kline 2-sphere char­ac­ter­iz­a­tion [e2]. In the peri­od 1915–1920, Moore [e1] had de­veloped a to­po­lo­gic­al char­ac­ter­iz­a­tion of the 2-sphere. A space \( X \) is a 2-sphere if and only if \( X \) is a com­pact, con­nec­ted, loc­ally con­nec­ted met­ric space of more than one point such that

1. no point sep­ar­ates \( X \),
2. \( X \) sat­is­fies the Jordan Curve The­or­em, and
3. each point has ar­bit­rar­ily small neigh­bor­hoods bounded by simple closed curves.

“J. R. Kline, Moore’s first Ph.D. stu­dent, con­jec­tured that the three con­di­tions above could be re­placed by

1. no point sep­ar­ates \( X \), and
2. each simple closed curve in \( X \) sep­ar­ates \( X \).

“Bing planned to sub­mit his pa­per to the Bul­let­in of the A.M.S. He was go­ing to an A.M.S. meet­ing shortly and found out that the ed­it­or to whom he would sub­mit the pa­per would be at the meet­ing. He took his pa­per with him and handed it to the ed­it­or, say­ing he wanted to sub­mit it to the Bul­let­in. When the ed­it­or saw Bing’s res­ult he looked at Bing and said, ‘Have you had this pa­per read by a reput­able math­em­atician?’ Without miss­ing a beat, Bing replied, ‘Yes, I have read it.’

“I can still re­mem­ber the smile on Dr. Moore’s face and the twinkle in his eyes as he fin­ished telling this story.”

“I first met R H Bing in 1949 when I was an un­der­gradu­ate at the Uni­versity of Wis­con­sin and I took his course, To­po­logy of the Plane, which he gave in the meth­od of his thes­is ad­visor, R. L. Moore. Bing gave the stu­dents four or five pages be­gin­ning with defin­i­tions of open and closed sets, etc., and a list of state­ments about them, some of which were true and some false; the stu­dents were to prove them or prove them false. In each class he would give a few ex­amples and talk about the sub­ject but didn't prove any of the the­or­ems. Then, go­ing down the list, day to day, the stu­dents were to prove or dis­prove the state­ments, at the black­board. I re­call that the first prob­lem for the class was to show that the uni­on of two open sets was open, which, after two days of proofs con­sist­ing of as­ser­tions that it was true, I proved it. The hard­est one I did was that a con­tinu­ous func­tion on a closed in­ter­val is uni­formly con­tinu­ous. Four­teen years later, I met up with Bing at the In­sti­tute for Ad­vanced Study and he com­men­ted that on my per­form­ance in that class he was not sur­prised to find me a math­em­atician.”