Celebratio Mathematica

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

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

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

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

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

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

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

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

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

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 \( E^3 \).

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

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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R H Bing

R H Bing: 1914–1986

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