Celebratio Mathematica

[1]E. F. Beckenbach and R. H. Bing: “Conformal minimal varieties,” Duke Math. J. 10 : 4 (1943), pp. 637–640. MR 0009498 Zbl 0063.00271 article

[2]E. F. Beckenbach and R. H. Bing: “On generalized convex functions,” Trans. Am. Math. Soc. 58 : 2 (September 1945), pp. 220–230. MR 0013169 Zbl 0060.14908 article

[3]R. H. Bing: “Collections filling up a simple plane web,” Bull. Am. Math. Soc. 51 : 10 (1945), pp. 674–679. MR 0013300 Zbl 0060.40309 article

A web has been defined by R. L. Moore [1943]. A compact plane continuum \( W \) is said [Bing 1946] to be a simple plane web if there exist an upper semicontinuous collection \( G \) of mutually exclusive continua filling up \( W \) and another such collection \( H \) also filling up \( W \) such that

(\( 1{} \)) \( G \) is a dendron with respect to its elements and so is \( H \), and
(\( 2{} \)) if \( g \) and \( h \) are elements of \( G \) and \( H \) respectively, the common part of \( g \) and \( h \) exists and is totally disconnected.

It has been shown [Bing 1946] that we have an equivalent definition if we substitute for (\( 1{} \)) in the above definition the condition that

(\( 1^{\prime} \)) \( G \) is a dendron with respect to its elements and \( H \) is an arc with respect to its elements.

The present paper shows that an equivalent definition is obtained if condition (\( 1{} \)) is omitted.

[4]R. H. Bing: “Generalizations of two theorems of Janiszewski,” Bull. Am. Math. Soc. 51 : 12 (1945), pp. 954–960. Part II was published in Bull. Am. Math. Soc. 52:6 (1946). MR 0013904 Zbl 0060.40402 article

[5]R. H. Bing: Concerning simple plane webs. Ph.D. thesis, University of Texas at Austin, 1945. Advised by R. L. Moore. See also article of same title in Trans. Am. Math. Soc. 60:1 (1946). MR 2937660 phdthesis

[6]R. H. Bing: “Converse linearity conditions,” Am. J. Math. 68 : 2 (April 1946), pp. 309–318. MR 0015443 Zbl 0060.14402 article

[7]R. H. Bing: “Generalizations of two theorems of Janiszewski, II,” Bull. Am. Math. Soc. 52 : 6 (1946), pp. 478–480. Part I was published in Bull. Am. Math. Soc. 51:12 (1945). MR 0016647 article

[8]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

[9]R. H. Bing: “The Kline sphere characterization problem,” Bull. Am. Math. Soc. 52 : 8 (1946), pp. 644–653. MR 0016645 Zbl 0060.40501 article

[10]R. H. Bing: “Sets cutting the plane,” Ann. Math. (2) 47 : 3 (July 1946), pp. 476–479. MR 0016648 Zbl 0060.40403 article

[11]R. H. Bing: “Extending a metric,” Duke Math. J. 14 : 3 (1947), pp. 511–519. MR 0024609 Zbl 0030.08003 article

[12]R. H. Bing: “Skew sets,” Am. J. Math. 69 : 3 (July 1947), pp. 493–498. MR 0021685 Zbl 0033.40303 article

[13]R. H. Bing: “A homogeneous indecomposable plane continuum,” Duke Math. J. 15 : 3 (1948), pp. 729–742. MR 0027144 Zbl 0035.39103 article

[14]R. H. Bing: “Solution of a problem of R. L. Wilder,” Am. J. Math. 70 : 1 (January 1948), pp. 95–98. MR 0023529 Zbl 0035.10903 article

[15]R. H. Bing: “Some characterizations of arcs and simple closed curves,” Am. J. Math. 70 : 3 (July 1948), pp. 497–506. MR 0025722 Zbl 0041.31801 article

[16]R. H. Bing: “A convex metric for a locally connected continuum,” Bull. Am. Math. Soc. 55 : 12 (1949), pp. 812–819. MR 0031712 Zbl 0035.10801 article

[17]R. H. Bing: “Complementary domains of continuous curves,” Fund. Math. 36 : 1 (1949), pp. 303–318. MR 0038063 Zbl 0039.39501 article

[18]R. H. Bing: “Partitioning a set,” Bull. Am. Math. Soc. 55 : 12 (1949), pp. 1101–1110. MR 0035429 Zbl 0036.11702 article

[19]R. H. Bing and E. E. Floyd: “Coverings with connected intersections,” Trans. Am. Math. Soc. 69 : 3 (November 1950), pp. 387–391. MR 0043453 Zbl 0039.39404 article

[20]R. H. Bing: “An equilateral distance,” Am. Math. Mon. 58 : 6 (June–July 1951), pp. 380–383. MR 0042725 Zbl 0043.16702 article

[21]R. H. Bing: “Metrization of topological spaces,” Canadian J. Math. 3 (1951), pp. 175–186. MR 0043449 Zbl 0042.41301 article

[22]R. H. Bing: “Concerning hereditarily indecomposable continua,” Pacific J. Math. 1 : 1 (1951), pp. 43–51. MR 0043451 Zbl 0043.16803 article

[23]R. H. Bing: “Higher-dimensional hereditarily indecomposable continua,” Trans. Am. Math. Soc. 71 : 2 (1951), pp. 267–273. MR 0043452 Zbl 0043.16901 article

[24]R. H. Bing: “Snake-like continua,” Duke Math. J. 18 : 3 (1951), pp. 653–663. MR 0043450 Zbl 0043.16804 article

[25]R. H. Bing: “A characterization of 3-space by partitionings,” Trans. Am. Math. Soc. 70 : 1 (January 1951), pp. 15–27. MR 0044827 Zbl 0042.41903 article

[26]R. H. Bing: “Partitioning continuous curves,” Bull. Am. Math. Soc. 58 : 5 (1952), pp. 536–556. MR 0049550 Zbl 0048.41203 article

[27]R. H. Bing: “The place of topology in a teacher training program” in Symposium on teacher education in mathematics (Madison, WI, 26–30 August 1952). 1952. 4 pages. incollection

[28]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

[29]R. H. Bing: “Book review: ‘General topology’,” Bull. Am. Math. Soc. 59 : 4 (1953), pp. 410. Book by W. Sierpiński (University of Toronto Press, 1952). MR 1565503 article

[30]R. H. Bing: “A connected countable Hausdorff space,” Proc. Am. Math. Soc. 4 : 3 (1953), pp. 474. MR 0060806 Zbl 0051.13902 article

[31]R. H. Bing: “Examples and counterexamples,” Pi Mu Epsilon J. 1 (1953), pp. 311–317. MR 0053499 article

[32]R. H. Bing: “A convex metric with unique segments,” Proc. Am. Math. Soc. 4 : 1 (1953), pp. 167–174. MR 0052763 Zbl 0050.38503 article

[33]R. H. Bing: “Locally tame sets are tame,” Ann. Math. (2) 59 : 1 (January 1954), pp. 145–158. MR 0061377 Zbl 0055.16802 article

[34]R. H. Bing: “Partially continuous decompositions,” Proc. Am. Math. Soc. 6 : 1 (1955), pp. 124–133. MR 0071003 Zbl 0064.16904 article

[35]R. H. Bing: “Some monotone decompositions of a cube,” Ann. Math. (2) 61 : 2 (March 1955), pp. 279–288. MR 0068207 Zbl 0064.41501 article

[36]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

[37]R. H. Bing: “What topology is here to stay?,” pp. 25–27 in Summer institute on set theoretic topology (Madison, WI, 24 July–20 August 1955). University of Wisconsin, 1957. In “Summary of lectures and seminars,” revised 1957. incollection

[38]R. H. Bing: “Decompositions of \( E^3 \) into points and tame arcs,” pp. 41–48 in Summer institute on set theoretic topology (Madison, WI, 24 July–20 August 1955). University of Wisconsin, 1957. In “Summary of lectures and seminars,” revised 1957. incollection

[39]R. H. Bing: “Point set topology,” pp. 306–335 in Insights into modern mathematics. Yearbook 23. National Council of Teachers of Mathematics (Washington, DC), 1957. incollection

[40]R. H. Bing: “Approximating surfaces with polyhedral ones,” pp. 49–53 in Summer institute on set theoretic topology (Madison, WI, 24 July–20 August 1955). American Mathematical Society (Providence, RI), 1957. In “Summary of lectures and seminars,” revised 1957. An expanded version of this article was published in Ann. Math. 65:3 (1957). incollection

[41]R. H. Bing: “The pseudo-arc,” pp. 72–57 in Summer institute on set theoretic topology (Madison, WI, 24 July–20 August 1955). University of Wisconsin, 1957. In “Summary of lectures and seminars,” revised 1957. incollection

[42]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

[43]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

We find that a useful theorem for \( E^2 \) does not generalize to \( E^3 \). R. L. Moore showed (Theorem 22 of [1925]) that if \( G \) is an upper semicontinuous decomposition of \( E^2 \) such that the elements of \( G \) are bounded continua which do not separate \( E^2 \), then the decomposition space is topologically equivalent to \( E^2 \). One might wonder if a decomposition space \( G \) is topologically equivalent to \( E^3 \) in the case where \( G \) is an upper semicontinuous decomposition of \( E^3 \) such that the complement of each element of \( G \) is topologically equivalent to the complement of a point in \( E^3 \). In this paper we show that there are such decompositions of \( E^3 \) whose decomposition spaces are not topologically \( E^3 \).

[44]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

[45]R. H. Bing: “The cartesian product of a certain nonmanifold and a line is \( E_4 \),” Bull. Am. Math. Soc. 64 : 3 (1958), pp. 82–84. An expanded version of this article was published in Ann. Math. 70:3 (1959). MR 0097034 Zbl 0084.19102 article

[46]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

[47]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

It was shown in Theorem 1 of [Bing 1957] that for each topological 2-sphere \( S \) in \( E^3 \) and each positive number \( \varepsilon \) there is a polyhedral 2-sphere \( S^{\prime} \) and a homeomorphism \( h \) of \( S \) onto \( S^{\prime} \) such that \( h \) moves no point by more than \( \varepsilon \). This result was extended [Bing 1957, Theorems 7, 8] to show that topological surfaces in triangulated 3-manifolds can be approximated by polyhedral surfaces.

The main purpose of this paper is to extend the above mentioned results and show that if in \( E^3 \) (or any other triangulated 3-manifold with boundary), \( P \) is a closed set which is the topological image of a 2-complex (finite or infinite) and \( f(x) \) is a positive continuous function defined on \( P \), then there is a homeomorphism \( h \) of \( P \) onto a polyhedron \( P^{\prime} \) such that \( h \) does not move any point \( x \) of \( P \) by as much as \( f(x) \). Three applications of this approximation theorem for 2-complexes are then given.

[48]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

[49]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

[50]R. H. Bing: Geometry, mathematics for high school. Technical report, School Mathematics Study Group, 1959. techreport

[51]R. H. Bing: “Each homogeneous nondegenerate chainable continuum is a pseudo-arc,” Proc. Am. Math. Soc. 10 : 3 (June 1959), pp. 345–346. MR 0105072 Zbl 0105.16701 article

[52]R. H. Bing and F. B. Jones: “Another homogeneous plane continuum,” Trans. Am. Math. Soc. 90 : 1 (1959), pp. 171–192. MR 0100823 Zbl 0084.18903 article

[53]R. H. Bing and M. L. Curtis: “Imbedding decompositions of \( E^3 \) in \( E^4 \),” Proc. Am. Math. Soc. 11 : 1 (February 1960), pp. 149–155. MR 0117692 Zbl 0119.18804 article

[54]R. H. Bing: Elementary point set topology, published as Am. Math. Mon. 67 : 7, part 2 (1960). Number 8 of the Herbert Ellsworth Slaught Memorial Papers. MR 0123286 Zbl 0095.37601 book

[55]R. H. Bing: “A simple closed curve is the only homogeneous bounded plane continuum that contains an arc,” Canad. J. Math. 12 (1960), pp. 209–230. MR 0111001 Zbl 0091.36204 article

[56]R. H. Bing: “Tame Cantor sets in \( E^3 \),” Pacific J. Math. 11 : 2 (1961), pp. 435–446. MR 0130679 Zbl 0111.18606 article

[57]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

[58]R. H. Bing and N. D. Kazarinoff: “On the finiteness of the number of reflections that change a nonconvex plane polygon into a convex one,” Mat. Prosvesh. 6 (1961), pp. 205–207. In Russian. article

[59]R. H. Bing: “A set is a 3 cell if its cartesian product with an arc is a 4 cell,” Proc. Am. Math. Soc. 12 : 1 (February 1961), pp. 13–19. MR 0123303 Zbl 0111.18702 article

[60]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

[61]R. H. Bing: “Point-like decompositions of \( E^3 \),” Fund. Math. 50 : 4 (1961/1962), pp. 431–453. MR 0137104 Zbl 0109.15503 article

An upper semicontinuous decomposition \( G \) of \( E^n \) (Euclidean \( n \) space) is point-like if for each element \( g \) of \( G \), \( E^n - g \) is topologically equivalent to the complement of a point. We call such an upper semicontinuous decomposition a point-like decomposition.

If \( G \) is a point-like decomposition of \( E^2 \) (or \( E^1 \)), the resulting decomposition space is topologically \( E^2 \) (or \( E^1 \)) [Moore 1925]. However, there are examples of point-like decompositions of \( E^3 \) whose decomposition spaces are topologically different from \( E^3 \). We give another such example in this paper and suggest a decomposition which may be an example.

[62]R. H. Bing: “Decompositions of \( E^3 \),” pp. 5–21 in Topology of 3-manifolds and related topics (Athens, GA, 14 August–8 September 1961). Edited by M. K. Fort, Jr. Prentice-Hall (Englewood Cliffs, NJ), 1962. MR 0141088 incollection

[63]R. H. Bing: “Embedding circle-like continua in the plane,” Canad. J. Math. 14 (1962), pp. 113–128. MR 0131865 Zbl 0101.15403 article

[64]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

[65]R. H. Bing: “Applications of the side approximation theorem for surfaces,” pp. 91–95 in General topology and its relations to modern analysis and algebra (Prague, September 1961). Academia (Prague), 1962. MR 0154266 Zbl 0112.38603 incollection

[66]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

[67]R. H. Bing: “Approximating surfaces from the side,” Ann. Math. (2) 77 : 1 (January 1963), pp. 145–192. MR 0150744 Zbl 0115.40603 article

[68]R. H. Bing: “Correction to ‘Necessary and sufficient conditions that a 3-manifold be \( S^3 \)’,” Ann. Math. (2) 77 : 1 (January 1963), pp. 210. Correction to article in Ann. Math. 68:1 (1958). MR 0142115 Zbl 0115.17304 article

[69]R. H. Bing: “Embedding surfaces in 3-manifolds,” pp. 457–458 in Proceedings of the International Congress of Mathematicians (Stockholm, 15–22 August 1962), vol. 1. Institute Mittag-Leffler (Djursholm), 1963. MR 0176455 Zbl 0137.17805 incollection

Is a 2-sphere in \( E^3 \) tame if it can be pierced at each point by a straight line interval? Is it tame if it can be pierced along each arc in it by a tame disk? Is a 2-sphere in \( E^3 \) tame if it is homogeneous under a space homeomorphism? Is a 2-sphere in \( E^3 \) tame if each Sierpiński curve in it can be pushed to either side? Is it tame if it can be approximated from either side by a singular 2-sphere? Does one obtain \( E^3 \) if the interior of a wild 2-sphere is replaced by the interior of a tame one? What restrictions imply that a surface is tamely embedded in a 3-manifold? These are samples of some of the interesting unsolved problems about the embedding of surfaces in 3-manifolds.

[70]R. H. Bing: “Retractions onto spheres,” Am. Math. Mon. 71 : 5 (May 1964), pp. 481–484. MR 0162236 Zbl 0117.40604 article

Suppose that \( S \) is a 2-sphere in \( E^3 \). It is known that \( S \) separates \( E^3 \) into exactly two components and is the common boundary of each. We call the unbounded component \( \operatorname{Ext} S \) and the bounded component \( \operatorname{Int} S \).

In case \( S \) is round, it is obvious that there is a retraction of \( S+\operatorname{Ext} S \) onto \( S \). One which comes quickly to mind sends a point \( p \) of \( S+\operatorname{Ext} S \) to the point where the segment from \( p \) to the center of \( \operatorname{Int} S \) intersects \( S \). If the sphere is wild, however, as the Alexander horned sphere illustrated on page 176 of [Hocking and Young 1961], it is not so obvious that there is such a retraction. It is the purpose of this note to show that there is.

[71]R. H. Bing and V. L. Klee: “Every simple closed curve in \( E^3 \) is unknotted in \( E^4 \),” J. London Math. Soc. 39 : 1 (1964), pp. 86–94. MR 0161336 Zbl 0116.40703 article

[72]R. H. Bing: “Pushing a 2-sphere into its complement,” Michigan Math. J. 11 : 1 (1964), pp. 33–45. MR 0160194 Zbl 0117.17101 article

[73]R. H. Bing and K. Borsuk: “A 3-dimensional absolute retract which does not contain any disk,” Fund. Math. 54 : 2 (1964), pp. 159–175. MR 0161312 Zbl 0118.18002 article

[74]R. H. Bing: “The simple connectivity of the sum of two disks,” Pacific J. Math. 14 : 2 (1964), pp. 439–455. MR 0164328 Zbl 0117.40701 article

[75]R. H. Bing and J. M. Kister: “Taming complexes in hyperplanes,” Duke Math. J. 31 : 3 (1964), pp. 491–511. MR 0164329 Zbl 0124.16701 article

In this paper we investigate conditions that suffice for an imbedding of a complex in a Euclidean space to be tame. We also consider the more general question of when two imbeddings are equivalent. We show, for example, with a dimension restriction that if the image under an imbedding of a \( k \)-complex is contained in a hyperplane of codimension \( k \), then the imbedding is tame. More precisely we prove:

Let \( K \) be a finite \( k \)-dimensional complex and \( h \) an imbedding of \( K \) into an \( n \)-plane \( E^n \) in \( E^{n+k} \), where \( k+2\leq n \). Let \( \varepsilon \) be any positive number. Then there exists an isotopy \( G_t \) (\( t\in I \)) of \( E^{n+k} \) onto itself such that

\( g_0 \) is the identity,
\( g_1 h \) is piecewise linear,
\( g_t \) is the identity outside an \( \varepsilon \)-neighborhood of \( h(K) \) for each \( t \) in \( I \),
each point of \( E^{n+k} \) moves along a polygonal path under \( g_t \) (\( t\in I \)) having length less than \( \varepsilon \).

[76]R. H. Bing: “Spheres in \( E^3 \),” Am. Math. Mon. 71 : 4 (April 1964), pp. 353–364. MR 0165507 Zbl 0116.40605 article

[77]R. H. Bing and A. Kirkor: “An arc is tame in 3-space if and only if it is strongly cellular,” Fund. Math. 55 : 2 (1964), pp. 175–180. MR 0170330 Zbl 0129.15902 article

[78]R. H. Bing: “Some aspects of the topology of 3-manifolds related to the Poincaré conjecture,” pp. 93–128 in Lectures on modern mathematics, vol. 2. Edited by T. L. Saaty. Wiley (New York), 1964. MR 0172254 Zbl 0126.39104 incollection

[79]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

[80]R. H. Bing: “A translation of the normal Moore space conjecture,” Proc. Am. Math. Soc. 16 : 4 (1965), pp. 612–619. MR 0181976 Zbl 0134.40906 article

[81]R. H. Bing: Computing the fundamental group of the complements. Technical report 2, Washington State University, 1965. techreport

[82]R. H. Bing: “Improving the side approximation theorem,” Trans. Am. Math. Soc. 116 (1965), pp. 511–525. MR 0192479 Zbl 0129.39701 article

Side Approximation Theorem for 2-Manifolds. Suppose \( M^2 \) is a connected 2-manifold (perhaps noncompact) in a connected 3-manifold \( M^3 \) such that \( M^3 - M^2 = U_1 + U_2 \) (mutually separated) and \( f \) is a positive continuous function defined on \( M^2 \). Then there is a homeomorphism \( h: M^2 \times [-1,1]\to M^3 \) such that

each \( h_t(M^2) \) is tame,
\( D(m,h_t(m)) < f(m) \) \( (m\in M^2 \), \( -1\leq t\leq 1) \),
\( M^2 - h(M^2\times (-1,1)) \) is covered by the interiors of a locally finite collection of mutually exclusive disks in \( M^2 \) such that the diameter of each is less than the minimum value of \( f \) on it,
for \( 0 < t \leq 1 \), \( \overline{U}_1\cdot h_{-t}(M^2) \) is covered by the interiors of a locally finite collection of mutually exclusive disks in \( h_t(M^2) \) each of diameter less than \( f(x) \) if \( h_t(x) \) lies in the disk, and
for \( 0 < t\leq 1 \), \( \overline{U}_2\cdot h_{-t}(M^2) \) is covered by the interiors of a locally finite collection of mutually exclusive disks in \( h_{-t}(M^2) \) each of diameter less than \( f(x) \) if \( h_{-t}(x) \) lies in the disk.

In fact, if \( {M^2}^{\prime} \) is a closed subset of \( M^2 \) which is a tame 2-manifold with boundary, \( h \) can be chosen so that \( h_0 \) is the identity on \( {M^2}^{\prime} \).

[83]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

[84]R. H. Bing: “Mapping a 3-sphere onto a homotopy 3-sphere,” pp. 89–99 in Topology seminar (Madison, WI, summer 1965). Edited by R. H. Bing and R. J. Bean. Annals of Mathematics Studies 60. Princeton University Press, 1966. MR 0219071 Zbl 0152.22603 incollection

[85]Topology seminar (Madison, WI, summer 1965). Edited by R. H. Bing and R. J. Bean. Annals of Mathematics Studies 60. Princeton University Press, 1966. MR 0202100 Zbl 0151.00208 book

[86]R. H. Bing: “A hereditarily infinite dimensional space,” pp. 56–62 in General topology and its relations to modern analysis and algebra (Prague, 30 August–4 September 1966), vol. II. Edited by J. Novák. Academia (Prague), 1967. MR 0233336 Zbl 0162.54802 incollection

[87]S. Armentrout and R. H. Bing: “A toroidal decomposition of \( E^3 \),” Fund. Math. 60 : 1 (1967), pp. 81–87. MR 0206925 Zbl 0145.19702 article

An upper semi-continuous decomposition \( G \) of \( E^3 \) is defined to be a toroidal decomposition if and only if the following condition holds: There is a sequence \( M_1 \), \( M_2 \), \( M_3 \), \( \dots \) of compact 3-manifolds-with-boundary in \( E^3 \) such that (1) for each \( i \), \( M_{i+1}\subset \operatorname{Int} M_i \) and each component of \( M_i \) is a solid torus (cube with one handle) and (2) \( A \) is a non-degenerate element of \( G \) if and only if \( A \) is a non-degenerate component of \( M_1\cdot M_2\cdot M_3\dots \).

Bing proved in [1952] that the union of two solid Alexander horned spheres, sewed together along their boundaries, was homeomorphic to \( S^3 \). A major step in this proof consists of showing that for a certain toroidal decomposition \( H \) of \( E^3 \) into tame arcs and one-point sets, the decomposition space associated with \( H \) is homeomorphic to \( E^3 \). Keldyš raised the following question in [1961]:

Does every toroidal decomposition of \( E^3 \) into tame arcs and one-point sets yield \( E^3 \) as its decomposition space?

In this paper we give a negative answer to this question.

[88]R. H. Bing: “Improving the intersections of lines and surfaces,” Michigan Math. J. 14 : 2 (1967), pp. 155–159. MR 0206927 Zbl 0149.21001 article

[89]R. H. Bing: “Challenging conjectures,” Am. Math. Mon. 74 : 1, part 2 (January 1967), pp. 56–64. MR 0203661 Zbl 0153.24101 article

[90]R. H. Bing: “Radial engulfing,” pp. 1–18 in Conference on the topology of manifolds (Michigan State University, East Lansing, MI, 1967). Edited by J. G. Hocking. Prindle, Weber & Schmidt (Boston), 1968. MR 0238284 Zbl 0186.57505 incollection

[91]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

[92]R. H. Bing: “The elusive fixed point property,” Am. Math. Mon. 76 : 2 (February 1969), pp. 119–132. MR 0236908 Zbl 0174.25902 article

[93]R. H. Bing: “Extending monotone decompositions of 3-manifolds,” pp. 7–35 in Proceedings of the Auburn topology conference (Auburn, AL, 13–15 March 1969). Edited by W. R. R. Transue. Auburn University, 1969. Dedicated to F. Burton Jones on the occasion of his 60th birthday. See also article in Trans. Am. Math. Soc. 149:2 (1970). MR 0370592 Zbl 0257.57002 incollection

Floyd Burton Jones	Related
William Raoul Reagle Transue	Related

[94]R. H. Bing: “Retractions onto ANR’s,” Proc. Am. Math. Soc. 21 : 3 (June 1969), pp. 618–620. MR 0239583 Zbl 0176.20102 article

[95]R. H. Bing: “Extending monotone decompositions of 3-manifolds,” Trans. Am. Math. Soc. 149 : 2 (1970), pp. 351–369. See also article in Proceedings of the Auburn topology conference (1969). MR 0263051 Zbl 0205.53401 article

[96]R. H. Bing: Topology of 3-manifolds (Laramie, WY, 24–28 August 1970). American Mathematical Society (Providence, RI), 1970. Notes distributed in conjunction with the Colloquium Lectures at the seventy-fifth summer meeting of the AMS. booklet

[97]R. H. Bing: “The monotone mapping problem,” pp. 99–115 in Topology of manifolds (University of Georgia, Athens, GA, 11–22 August 1969). Edited by J. C. Cantrell and C. H. Edwards. Markham Mathematics Series. Markham (Chicago), 1970. MR 0275379 Zbl 0283.57004 incollection

It is shown that for \( m = 3,4,\dots \) there is a monotone map of Euclidean \( n \)-space \( E^n \) onto itself that is not compact. This completes the monotone mapping theorem posed by G. T. Whyburn. A key lemma in the treatment shows that there is a monotone map of a cube \( I^2 \) onto itself such that each point inverse intersects a base \( I^2 \) of \( I^3 \). If \( f \) is a map of \( I^3 \) onto \( I^3 \) which is a homeomorphism on \( \operatorname{Int}I^3 \) and takes \( I^2 \) homeomorphically into \( I^2 \), one calls \[ f(\operatorname{Int}I^2\cup\operatorname{Int}I^3) \] a drainage system for \( I^3 \). It is shown that there is a drainage system \( f(\operatorname{Int}I^2 \) \( \cup\operatorname{Int}I^3) \) for \( I^3 \) and a monotone map \( g \) of \[ I^3 - f(\operatorname{Int}I^2\cup\operatorname{Int}I^3) \] onto \( I^3 \) such that \( g \) is the identity on \( \operatorname{Bd}I^3 - \operatorname{Int}I^2 \).

James Cecil Cantrell	Related
C. H. Edwards	Related

[98]R. H. Bing and J. M. Martin: “Cubes with knotted holes,” Trans. Am. Math. Soc. 155 : 1 (1971), pp. 217–231. MR 0278287 Zbl 0213.25005 article

[99]R. H. Bing and J. M. Martin: “Monotone images of \( E^3 \),” pp. 55–77 in The proceedings of the first conference on monotone mappings and open mappings (SUNY, Binghamton, NY, 8–11 October 1970). Edited by L. F. McAuley. SUNY Binghamton, 1971. MR 0281175 Zbl 0239.57003 incollection

Louis F. McAuley	Related
Joseph Morison Martin	Related

[100]R. H. Bing: “Models for \( S^3 \),” pp. 1–31 in Visiting scholars’ lectures (Texas Tech University, Lubbock, TX, 1970–1971). Edited by G. L. Baldwin and J. D. Tarwater. Mathematics Series 9. Texas Tech University (Lubbock, TX), 1971. MR 0358782 Zbl 0239.57004 incollection

George L. Baldwin	Related
J. Dalton Tarwater	Related

[101]R. H. Bing: “Award for distinguished service to Professor Raymond L. Wilder,” Am. Math. Mon. 80 : 2 (February 1973), pp. 117–119. MR 1536974 article

[102]R. H. Bing, W. W. Bledsoe, and R. D. Mauldin: “Sets generated by rectangles,” Pacific J. Math. 51 : 1 (1974), pp. 27–36. MR 0357124 Zbl 0261.04001 article

For any family \( F \) of sets, let \( \mathcal{B}(F) \) denote the smallest \( \sigma \)-algebra containing \( F \). Throughout this paper \( X \) denotes a set and \( \mathcal{R} \) the family of sets of the form \( A\times B \), for \( A\subseteq X \) and \( B\subseteq X \). It is of interest to find conditions under which the following holds:

(1) Each subset of \( X\times X \) is a member of \( \mathcal{B}(\mathcal{R}) \).

The interesting case is when \( \omega_t < \operatorname{Card} X \leq c \), since results for other cases are known.

It is shown in Theorem 9 that (1) is equivalent to

(2) There is a countable ordinal a such that each subset of \( X\times X \) can be generated from \( \mathcal{R} \) in \( \alpha \) Baire process steps.

It is also shown that the two-dimensional statements (1) and (2) are equivalent to the one-dimensional statement

(3) There is a countable ordinal \( \alpha \) such that for each family \( H \) of subsets of \( X \) with \( \operatorname{Card} H = \operatorname{Card} X \), there is a countable family \( G \) such that each member of \( H \) can be generated from \( G \) in \( \alpha \) steps.

It is shown in Theorem 5 that the continuum hypothesis (CH) is equivalent to certain statements about rectangles of the form (1) and (2) with \( \alpha = 2 \).

Woodrow Wilson Bledsoe	Related
Richard Daniel Mauldin	Related

[103]R. H. Bing: “An unusual map of a 3-cell onto itself,” pp. 15–33 in Topology conference (Virginia Polytechnic Institute and State University, Blacksburg, VA, 22–24 March 1973). Edited by H. F. Dickman, Jr. and P. Fletcher. Lecture Notes in Mathematics 375. Springer (Berlin), 1974. MR 0358783 Zbl 0291.57004 incollection

H. F. Dickman, Jr	Related
Peter Fletcher	Related

[104]R. H. Bing: “Topology, general,” pp. 509–514 in Encyclopædia Britannica, vol. 18. Encyclopædia Britannica (Chicago), 1974. incollection

[105]R. H. Bing: “Topologia conjuntista,” pp. 126–132 in Universitas enciclopedia temática, vol. X. Salvat (Barcelona), 1974. incollection

[106]R. H. Bing: “Vertical general position,” pp. 16–41 in Geometric topology (Park City, UT, 19–24 February 1974). Edited by L. C. Glaser and T. B. Rushing. Lecture Notes in Mathematics 438. Springer (Berlin), 1975. MR 0394685 Zbl 0331.57008 incollection

T. Benny Rushing	Related
Leslie C. Glaser	Related

[107]R. H. Bing: “Pulling back feelers,” pp. 245–266 in Convegno sulla topologia insiemistica e generale [Conference on point-set and general topology] (INDAM, Rome, 1973). Symposia Mathematica XVI. Academic Press (London), 1975. MR 0400231 Zbl 0322.57001 incollection

[108]R. H. Bing: “Set theory,” pp. 254–255 in McGraw-Hill encyclopedia of science and technology, 4th edition. McGraw-Hill (New York), 1977. incollection

[109]R. H. Bing and M. Starbird: “Super triangulations,” Pacific J. Math. 74 : 2 (1978), pp. 307–325. MR 0478170 Zbl 0384.57005 article

[110]R. H. Bing and M. Starbird: “Linear isotopies in \( E^2 \),” Trans. Am. Math. Soc. 237 (March 1978), pp. 205–222. MR 0461510 Zbl 0397.57018 article

This paper deals with continuous families of linear embeddings (called linear isotopies) of finite complexes in the Euclidean plane \( E^2 \). Suppose \( f \) and \( g \) are two linear embeddings of a finite complex \( P \) with triangulation \( T \) into a simply connected open subset \( U \) of \( E^2 \) so that there is an orientation preserving homeomorphism \( H \) of \( E^2 \) to itself with \( H\circ f = g \). It is shown that there is a continuous family of embeddings \( h_t:P\to U \) (\( t\in [0,1] \)) so that \( h_0=f \), \( h_1=g \), and for each \( t \), \( h_t \) is linear with respect to \( T \).

It is also shown that if \( P \) is a PL star-like disk in \( E^2 \) with a triangulation \( T \) which has no spanning edges and \( f \) is a homeomorphism of \( P \) which is the identity on \( \operatorname{Bd} P \) and is linear with respect to \( T \), then there is an continuous family of homeomorphisms \( h_t:P\to P \) (\( t\in [0,1] \)) such that \( h_0=\mathrm{id} \), \( h_1=f \), and for each \( t \), \( h_t \) is linear with respect to \( T \). An example shows the necessity of the “star-like” requirement. A consequence of this last theorem is a linear isotopy version of the Alexander isotopy theorem — namely, if \( f \) and \( g \) are two PL embeddings of a disk \( P \) into \( E^2 \) so that \( f|_{\operatorname{Bd} P} = g|_{\operatorname{Bd} P} \), then there is a linear isotopy with respect to some triangulation of \( P \) which starts at \( f \), ends at \( g \), and leaves the boundary fixed throughout.

[111]R. H. Bing: “Award for distinguished service to Professor R. D. Anderson,” Am. Math. Mon. 85 : 2 (February 1978), pp. 73–74. article

[112]R. H. Bing and M. Starbird: “A decomposition of \( S^3 \) with a null sequence of cellular arcs,” pp. 3–21 in Geometric topology (Athens, GA, 1–12 August 1977). Edited by J. C. Cantrell. Academic Press (New York), 1979. MR 537722 Zbl 0479.57004 incollection

Michael Peter Starbird	Related
James Cecil Cantrell	Related

[113]R. H. Bing: “Sets in \( R^3 \) with the two-disk property,” pp. 43–44 in Proceedings of international conference on geometric topology (Institute of Mathematics, Warsaw, 24 August–2 September 1978). Edited by K. Borsuk and A. Kirkor. Polish Scientific Publishers (Warsaw), 1980. Zbl 0463.57002 incollection

Andrzej Kirkor	Related
Karol Borsuk	Related

[114]R. H. Bing: “Metrization problems,” pp. 3–16 in General topology and modern analysis (University of California, Riverside, CA, 28–31 May 1980). Edited by L. F. McAuley and M. M. Rao. Academic Press (New York), 1981. MR 619024 Zbl 0527.54027 incollection

Louis F. McAuley	Related
Malempati Madhusudana Rao	Related

[115]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

[116]Continua, decompositions, manifolds (University of Texas, Austin, TX, summer 1980). Edited by R. H. Bing, W. T. Eaton, and M. P. Starbird. University of Texas Press (Austin, TX), 1983. MR 711973 book

Michael Peter Starbird	Related
William T. Eaton	Related

[117]R. H. Bing: “Wild surfaces have some nice properties,” Michigan Math. J. 33 : 3 (1986), pp. 403–415. MR 856532 Zbl 0609.57005 article

[118]R. H. Bing: “Decompositions that destroy simple connectivity,” Illinois J. Math. 30 : 4 (1986), pp. 527–535. MR 857209 Zbl 0608.57012 article

[119]S. Singh: “R. H. Bing (1914–1986): A tribute,” pp. 5–8 in Special volume in honor of R. H. Bing (1914–1986), published as Topology Appl. 24 : 1–3 (1986). MR 872474 incollection

[120]Special volume in honor of R. H. Bing (1914–1986), published as Topology Appl. 24 : 1–3. Issue edited by S. Singh and T. L. Thickstun. Elsevier Science B.V. (Amsterdam), 1986. MR 872473 book

Sukhjit Singh	Related
Thomas Lusk Thickstun	Related

[121]R. D. Anderson and C. E. Burgess: “R. H. Bing: October 20, 1914–April 28, 1986,” Notices Am. Math. Soc. 33 : 4 (1986), pp. 595–596. MR 847220 article

Richard D. Anderson	Related
Cecil Edmund Burgess	Related

[122]L. Whyburn: “R. H. Bing 1949–50,” pp. 177–180 in Proceedings of the 1987 topology conference (Birmingham, AL, 1987), published as Topology Proc. 12 : 1. Issue edited by G. Gruenhage, D. Bennett, and L. Mohler. Auburn University (Birmingham, AL), 1987. MR 951715 Zbl 0643.01015 incollection

Donna Bennett	Related
Gary Gruenhage	Related
Lee K. Mohler	Related
Lucille Whyburn	Related

[123]F. B. Jones: “R. H. Bing,” pp. 181–186 in Proceedings of the 1987 topology conference (Birmingham, AL, 1987), published as Topology Proc. 12 : 1. Issue edited by G. Gruenhage, D. Bennett, and L. Mohler. Auburn University (Birmingham, AL), 1987. MR 951716 Zbl 0643.01014 incollection

Donna Bennett	Related
Gary Gruenhage	Related
Floyd Burton Jones	Related
Lee K. Mohler	Related

[124]M. Brown: “The mathematical work of R. H. Bing,” pp. 1–25 in Proceedings of the 1987 topology conference (Birmingham, AL, 1987), published as Topology Proc. 12 : 1. Issue edited by G. Gruenhage, D. Bennett, and L. Mohler. Auburn University (Birmingham, AL), 1987. MR 951703 Zbl 0661.01017 incollection

Donna Bennett	Related
Gary Gruenhage	Related
Lee K. Mohler	Related
Morton Brown	Related	Volume

[125]S. Singh: “Publications of R. H. Bing classified by the year,” pp. 27–37 in Proceedings of the 1987 topology conference (Birmingham, AL, 1987), published as Topology Proc. 12 : 1. Issue edited by G. Gruenhage, D. Bennett, and L. Mohler. Auburn University (Birmingham, AL), 1987. MR 951704 Zbl 0643.01016 incollection

Donna Bennett	Related
Gary Gruenhage	Related
Lee K. Mohler	Related
Sukhjit Singh	Related

[126]R. H. Bing: “Shrinking without lengthening,” Topology 27 : 4 (1988), pp. 487–493. MR 976590 Zbl 0673.57011 article

[127]R. H. Bing: Collected papers. Edited by S. Singh, S. Armentrout, and R. J. Daverman. American Mathematical Society (Providence, RI), 1988. In two volumes. MR 950859 Zbl 0665.01012 book

Steve Armentrout	Related
Virendra Singh	Related
Robert J. Daverman	Related

[128]M. Starbird: “R. H. Bing’s human and mathematical vitality,” pp. 453–466 in Handbook of the history of general topology (San Antonio, TX, 1993), vol. 2. Edited by C. E. Aull and R. Lowen. History of Topology 2. Kluwer Academic (Dordrecht), 1998. MR 1795160 Zbl 0932.01042 incollection

Michael Peter Starbird	Related
Robert Lowen	Related
Charles E. Aull	Related

R H Bing

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