##
**The collected papers of R. H. Bing. Vol. 1 and 2. Ed. by Sukhjit Singh, Steve Armentrout, and Robert J. Daverman.**
*(English)*
Zbl 0665.01012

Providence, RI: American Mathematical Society (AMS). xix, 1654 p. $ 155.00 (1988).

Although he came to mathematical research rather late (Ph. D. at age 30), R. H. Bing was a prolific, prodigeous and powerful worker. In these two volumes all of his published research (some 100 papers) is reproduced. How wide spread these papers were is testified to by the fact that, in addition the American Mathematical Society, publication permission was required from 32 organizations. Seven of these were in foreign countries. In addition, the collection contains almost all of Bing’s other published work including survey papers, essays and Number 8 in the Mathematical Association’s Slaught Memorial Papers. Not included is Bing’s A.M.S. Colloquium publication (Vol. 40), Chapter X of “Insights into modern mathematics,” yearbook of National Council of Teachers of Mathematics (1957) and two or three minor items.

The collected papers of R. H. Bing are indexed in the table of contents and grouped under the following headings: I. General topology, II. Plane sets, III. Continua, IV. Topological classifications, V. Miscellaneous topics, and VI. Topology of 3-manifolds, the latter being all of volume 2. Subheadings under I include A. Metrization. In one of these eight papers is the theorem that every collection wise normal Moore space is metrizable, as well as his famous metrization theorem for Hausdorff spaces.

In IIA (Plane webs) is the difficult theorem from Bing’s Doctoral Thesis: A locally connected plane continuum is a simple web if and only if it remains connected and locally connected on the omission of any countable subset. He characterizes a plane web as a plane continuum which has to monotone decompositions such that each element of one intersects each element of the other in a (nonvoid) totally disconnected set.

In III, under subheadings A and C, Bing shows that the pseudo-arc is homogeneous, the circle of pseudo-arcs is homogeneous and the simple closed curve is the only homogeneous plane continuum that contains an arc. (It is still not known that these three are the only homogeneous plane continua.) In another paper in IIIA it is shown that hereditarily indecomposable continua of all dimensions exist. But in IIIB Bing proves some of his most powerful theorems: A locally connected continuum may be partitioned and every such continuum has a brick partitioning. Using these notions Bing settles a problem posed by Menger by showing that every locally connected continuum has a convex metric.

In IVA one finds the Kline sphere characterization: A nondegenerate, locally connected continuum which is separated by each of its simple closed curves, but by no pair of its points, is a 2-sphere. In IVB is a characterization of 3-space by partitionings. And in IVD, Bing and Anderson give a complete elementary proof that Hilbert space is the countable infinite product of lines.

In V (Miscellaneous topics) F, Bing solves a problem concerning connectedness proposed by Wilder in 1930. If M is the union of two plane sets each irreducibly connected from the point a to the point b but otherwise disjoint and M is locally connected, is M necessarily a simple closed curve? Bing answers this by decomposing the plane unit disk into connected sets each irreducible from the vertex a to the opposite vertex b which are disjoint except for a and b such that the union of any two of them is locally connected but not a simple closed curve.

Section VI is devoted to Bing’s papers on 3-manifolds (approximately 50 papers and 760 pages). So only a few of these results will be pointed out. The earliest of these is in VIB. Bing shows that if you sew two solid horned spheres together on their surfaces using the identity the result is a 3-sphere. The next major result was the construction of the “dog bone” space. This is a decomposition of \(E^ 3\) into points and tame arcs such that the decomposition space is different from \(E^ 3\). His alternative proof that every 3-manifold can be triangulated and the side approximation theorem laid the foundation for more than twenty years of research on 3-manifolds.

There are two more lists of Bing’s papers, one chronological and the other Bing’s arrangement by subject matter. Both are indexed. There is a list of Bing’s abstracts, a list of his Ph. D. students (35) and a chronology which gives the reader some inkling of what he did professionally besides research. (The beginning date of Bing’s employment at the University of Texas should be 1942, not 1943.) Reprinted from the Notices of the American Mathematical Society is a memorial to Bing by Anderson and Burgess.

These volumes begin with an article by S. Singh, “R. H. Bing, a study of his life”. Based on in-depth interviews, it is written as if Bing himself were talking to the reader. He talks about his life from childhood through graduate school. He discusses how some of his research came to be and philosophizes about teaching and other topics. He has a number of remarks about collaboration with others (he wrote papers with at least a dozen different mathematicians) and ends with the comment: “In general, I think better results are obtained by those who stand on their own feet and work alone.”

At one point in section 5 (Wisconsin years and mathematics of this time) there is some confusion. Bing reminisces, “When I was a graduate student at the University of Texas, I worked on the problem as to whether or not every normal Moore space is metrizable. This question had been called to Moise’s attention when he was visiting a meeting of the American Mathematical Society in New York.” It was the Souslin problem that Moise came back from New York with, not the normal Moore space problem. In the next paragraph Bing picks up the story of his work on the normal Moore space problem which lead to his metrization theorem for Hausdorff spaces. Bing continues, “It turned out later that Smirnoff from Russia and Nagata from Japan had gotten similar results about the same time. This suggests that when it is time for a new result to be born, that if one person does not get it, another person is likely to. No research mathematician is indispensible for progress in mathematical research.”

Certainly this work is indispensable to a mathematician working in 3- space topology and of great assistance to the mathematical historian working on the history of this century. Even the ordinary reader may find something of interest in Bing’s nontechnical papers. Bing, who often spoke of himself as a “salesman for mathematics,” says toward the end of Singh’s fascinating article, “I have a basic feeling that mathematics should be fun and should be fun for the participant.”

The collected papers of R. H. Bing are indexed in the table of contents and grouped under the following headings: I. General topology, II. Plane sets, III. Continua, IV. Topological classifications, V. Miscellaneous topics, and VI. Topology of 3-manifolds, the latter being all of volume 2. Subheadings under I include A. Metrization. In one of these eight papers is the theorem that every collection wise normal Moore space is metrizable, as well as his famous metrization theorem for Hausdorff spaces.

In IIA (Plane webs) is the difficult theorem from Bing’s Doctoral Thesis: A locally connected plane continuum is a simple web if and only if it remains connected and locally connected on the omission of any countable subset. He characterizes a plane web as a plane continuum which has to monotone decompositions such that each element of one intersects each element of the other in a (nonvoid) totally disconnected set.

In III, under subheadings A and C, Bing shows that the pseudo-arc is homogeneous, the circle of pseudo-arcs is homogeneous and the simple closed curve is the only homogeneous plane continuum that contains an arc. (It is still not known that these three are the only homogeneous plane continua.) In another paper in IIIA it is shown that hereditarily indecomposable continua of all dimensions exist. But in IIIB Bing proves some of his most powerful theorems: A locally connected continuum may be partitioned and every such continuum has a brick partitioning. Using these notions Bing settles a problem posed by Menger by showing that every locally connected continuum has a convex metric.

In IVA one finds the Kline sphere characterization: A nondegenerate, locally connected continuum which is separated by each of its simple closed curves, but by no pair of its points, is a 2-sphere. In IVB is a characterization of 3-space by partitionings. And in IVD, Bing and Anderson give a complete elementary proof that Hilbert space is the countable infinite product of lines.

In V (Miscellaneous topics) F, Bing solves a problem concerning connectedness proposed by Wilder in 1930. If M is the union of two plane sets each irreducibly connected from the point a to the point b but otherwise disjoint and M is locally connected, is M necessarily a simple closed curve? Bing answers this by decomposing the plane unit disk into connected sets each irreducible from the vertex a to the opposite vertex b which are disjoint except for a and b such that the union of any two of them is locally connected but not a simple closed curve.

Section VI is devoted to Bing’s papers on 3-manifolds (approximately 50 papers and 760 pages). So only a few of these results will be pointed out. The earliest of these is in VIB. Bing shows that if you sew two solid horned spheres together on their surfaces using the identity the result is a 3-sphere. The next major result was the construction of the “dog bone” space. This is a decomposition of \(E^ 3\) into points and tame arcs such that the decomposition space is different from \(E^ 3\). His alternative proof that every 3-manifold can be triangulated and the side approximation theorem laid the foundation for more than twenty years of research on 3-manifolds.

There are two more lists of Bing’s papers, one chronological and the other Bing’s arrangement by subject matter. Both are indexed. There is a list of Bing’s abstracts, a list of his Ph. D. students (35) and a chronology which gives the reader some inkling of what he did professionally besides research. (The beginning date of Bing’s employment at the University of Texas should be 1942, not 1943.) Reprinted from the Notices of the American Mathematical Society is a memorial to Bing by Anderson and Burgess.

These volumes begin with an article by S. Singh, “R. H. Bing, a study of his life”. Based on in-depth interviews, it is written as if Bing himself were talking to the reader. He talks about his life from childhood through graduate school. He discusses how some of his research came to be and philosophizes about teaching and other topics. He has a number of remarks about collaboration with others (he wrote papers with at least a dozen different mathematicians) and ends with the comment: “In general, I think better results are obtained by those who stand on their own feet and work alone.”

At one point in section 5 (Wisconsin years and mathematics of this time) there is some confusion. Bing reminisces, “When I was a graduate student at the University of Texas, I worked on the problem as to whether or not every normal Moore space is metrizable. This question had been called to Moise’s attention when he was visiting a meeting of the American Mathematical Society in New York.” It was the Souslin problem that Moise came back from New York with, not the normal Moore space problem. In the next paragraph Bing picks up the story of his work on the normal Moore space problem which lead to his metrization theorem for Hausdorff spaces. Bing continues, “It turned out later that Smirnoff from Russia and Nagata from Japan had gotten similar results about the same time. This suggests that when it is time for a new result to be born, that if one person does not get it, another person is likely to. No research mathematician is indispensible for progress in mathematical research.”

Certainly this work is indispensable to a mathematician working in 3- space topology and of great assistance to the mathematical historian working on the history of this century. Even the ordinary reader may find something of interest in Bing’s nontechnical papers. Bing, who often spoke of himself as a “salesman for mathematics,” says toward the end of Singh’s fascinating article, “I have a basic feeling that mathematics should be fun and should be fun for the participant.”

Reviewer: F.B.Jones

### MSC:

01A75 | Collected or selected works; reprintings or translations of classics |

54-03 | History of general topology |

01A70 | Biographies, obituaries, personalia, bibliographies |

57-03 | History of manifolds and cell complexes |