R. H. Bing :
“The Kline sphere characterization problem ,”
Bull. Am. Math. Soc.
52 : 8
(1946 ),
pp. 644–653 .
MR
0016645
Zbl
0060.40501
article
Abstract
BibTeX
The object of this paper is to give a solution to the following problem proposed by J. R. Kline: Is a nondegenerate, locally connected, compact continuum which is separated by each of its simple closed curves but by no pair of its points homeomorphic with the surface of a sphere? The answer is in the affirmative.
@article {key0016645m,
AUTHOR = {Bing, R. H.},
TITLE = {The {K}line sphere characterization
problem},
JOURNAL = {Bull. Am. Math. Soc.},
FJOURNAL = {Bulletin of the American Mathematical
Society},
VOLUME = {52},
NUMBER = {8},
YEAR = {1946},
PAGES = {644--653},
DOI = {10.1090/S0002-9904-1946-08614-0},
NOTE = {MR:0016645. Zbl:0060.40501.},
ISSN = {0002-9904},
}
R. H. Bing :
“A homogeneous indecomposable plane continuum ,”
Duke Math. J.
15 : 3
(1948 ),
pp. 729–742 .
MR
0027144
Zbl
0035.39103
article
Abstract
BibTeX
@article {key0027144m,
AUTHOR = {Bing, R. H.},
TITLE = {A homogeneous indecomposable plane continuum},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {15},
NUMBER = {3},
YEAR = {1948},
PAGES = {729--742},
DOI = {10.1215/S0012-7094-48-01563-4},
NOTE = {MR:0027144. Zbl:0035.39103.},
ISSN = {0012-7094},
}
R. H. Bing :
“Metrization of topological spaces ,”
Canadian J. Math.
3
(1951 ),
pp. 175–186 .
MR
0043449
Zbl
0042.41301
article
BibTeX
@article {key0043449m,
AUTHOR = {Bing, R. H.},
TITLE = {Metrization of topological spaces},
JOURNAL = {Canadian J. Math.},
FJOURNAL = {Canadian Journal of Mathematics},
VOLUME = {3},
YEAR = {1951},
PAGES = {175--186},
DOI = {10.4153/CJM-1951-022-3},
NOTE = {MR:0043449. Zbl:0042.41301.},
ISSN = {0008-414X},
}
R. H. Bing :
“Concerning hereditarily indecomposable continua ,”
Pacific J. Math.
1 : 1
(1951 ),
pp. 43–51 .
MR
0043451
Zbl
0043.16803
article
Abstract
BibTeX
@article {key0043451m,
AUTHOR = {Bing, R. H.},
TITLE = {Concerning hereditarily indecomposable
continua},
JOURNAL = {Pacific J. Math.},
FJOURNAL = {Pacific Journal of Mathematics},
VOLUME = {1},
NUMBER = {1},
YEAR = {1951},
PAGES = {43--51},
URL = {http://projecteuclid.org/euclid.pjm/1102613150},
NOTE = {MR:0043451. Zbl:0043.16803.},
ISSN = {0030-8730},
}
R. H. Bing :
“Locally tame sets are tame ,”
Ann. Math. (2)
59 : 1
(January 1954 ),
pp. 145–158 .
MR
0061377
Zbl
0055.16802
article
Abstract
BibTeX
@article {key0061377m,
AUTHOR = {Bing, R. H.},
TITLE = {Locally tame sets are tame},
JOURNAL = {Ann. Math. (2)},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {59},
NUMBER = {1},
MONTH = {January},
YEAR = {1954},
PAGES = {145--158},
DOI = {10.2307/1969836},
NOTE = {MR:0061377. Zbl:0055.16802.},
ISSN = {0003-486X},
}
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
Abstract
BibTeX
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 \( E^3 \) .
@article {key0092961m,
AUTHOR = {Bing, R. H.},
TITLE = {A decomposition of \$E^3\$ into points
and tame arcs such that the decomposition
space is topologically different from
\$E^3\$},
JOURNAL = {Ann. Math. (2)},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {65},
NUMBER = {3},
MONTH = {May},
YEAR = {1957},
PAGES = {484--500},
DOI = {10.2307/1970058},
NOTE = {MR:0092961. Zbl:0079.38806.},
ISSN = {0003-486X},
}
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
Abstract
BibTeX
@article {key0087090m,
AUTHOR = {Bing, R. H.},
TITLE = {Approximating surfaces with polyhedral
ones},
JOURNAL = {Ann. Math. (2)},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {65},
NUMBER = {3},
MONTH = {May},
YEAR = {1957},
PAGES = {465--483},
URL = {http://www.jstor.org/stable/1970057},
NOTE = {Expanded version of an article in \textit{Summer
institute on set theoretic topology}
(1957). MR:0087090. Zbl:0079.38805.},
ISSN = {0003-486X},
}
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
Abstract
BibTeX
A compact, connected 3-manifold \( M \) is topologically \( S^3 \) if each simple closed curve in \( M \) lies in a topological cube in \( M \) .
@article {key0095471m,
AUTHOR = {Bing, R. H.},
TITLE = {Necessary and sufficient conditions
that a 3-manifold be \$S^3\$},
JOURNAL = {Ann. Math. (2)},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {68},
NUMBER = {1},
MONTH = {July},
YEAR = {1958},
PAGES = {17--37},
DOI = {10.2307/1970041},
NOTE = {MR:0095471. Zbl:0081.39202.},
ISSN = {0003-486X},
}
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
Abstract
BibTeX
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.
@article {key0100841m,
AUTHOR = {Bing, R. H.},
TITLE = {An alternative proof that 3-manifolds
can be triangulated},
JOURNAL = {Ann. Math. (2)},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {69},
NUMBER = {1},
MONTH = {January},
YEAR = {1959},
PAGES = {37--65},
DOI = {10.2307/1970092},
NOTE = {MR:0100841. Zbl:0106.16604.},
ISSN = {0003-486X},
}
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
Abstract
BibTeX
@article {key0107228m,
AUTHOR = {Bing, R. H.},
TITLE = {The {C}artesian product of a certain
nonmanifold and a line is \$E^4\$},
JOURNAL = {Ann. Math. (2)},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {70},
NUMBER = {3},
MONTH = {November},
YEAR = {1959},
PAGES = {399--412},
DOI = {10.2307/1970322},
NOTE = {Expanded version of an article in \textit{Bull
Am. Math. Soc.} \textbf{64}:3 (1958).
MR:0107228. Zbl:0089.39501.},
ISSN = {0003-486X},
}
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
Abstract
People
BibTeX
In 1954, working independently, Bing and Jones each discovered a homogeneous plane continuum that was neither a simple closed curve nor a pseudo-arc. Neither knew of the others work until the titles of the papers appeared adjacent to each other on the 1954 summer program of the American Mathematical Society [Bing and Jones 1959; Jones 1955]. Inasmuch as both had discovered the same example, it was decided to make this a joint paper. The first part of this paper showing that the example — a circle of pseudo-arcs — is homogeneous was written by Bing. The latter part showing that such a circle of pseudo-arcs can be imbedded in the plane was prepared by Jones.
@article {key0100823m,
AUTHOR = {Bing, R. H. and Jones, F. B.},
TITLE = {Another homogeneous plane continuum},
JOURNAL = {Trans. Am. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical
Society},
VOLUME = {90},
NUMBER = {1},
YEAR = {1959},
PAGES = {171--192},
DOI = {10.2307/1993272},
NOTE = {MR:0100823. Zbl:0084.18903.},
ISSN = {0002-9947},
}
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
Abstract
BibTeX
@article {key0111001m,
AUTHOR = {Bing, R. H.},
TITLE = {A simple closed curve is the only homogeneous
bounded plane continuum that contains
an arc},
JOURNAL = {Canad. J. Math.},
FJOURNAL = {Canadian Journal of Mathematics},
VOLUME = {12},
YEAR = {1960},
PAGES = {209--230},
DOI = {10.4153/CJM-1960-018-x},
NOTE = {MR:0111001. Zbl:0091.36204.},
ISSN = {0008-414X},
}
R. H. Bing :
“Tame Cantor sets in \( E^3 \) ,”
Pacific J. Math.
11 : 2
(1961 ),
pp. 435–446 .
MR
0130679
Zbl
0111.18606
article
Abstract
BibTeX
Characterizations of tame Cantor sets are provided by Theorems 1.1, 3.1, 4.1 and 5.1. In §6 we prove theorems about the sums of tame Cantor sets and apply these results in §7 to show that for each closed 2-dimensional set \( X \) in \( E^3 \) there is a homeomorphism \( h \) of \( E^3 \) onto itself that is close to the identity and such that \( h(X) \) contains no straight line interval. An example is given of a disk containing intervals pointing in all directions showing that such a homeomorphism \( h \) may need to be something more than a rigid motion.
@article {key0130679m,
AUTHOR = {Bing, R. H.},
TITLE = {Tame {C}antor sets in \$E^3\$},
JOURNAL = {Pacific J. Math.},
FJOURNAL = {Pacific Journal of Mathematics},
VOLUME = {11},
NUMBER = {2},
YEAR = {1961},
PAGES = {435--446},
URL = {http://projecteuclid.org/euclid.pjm/1103037324},
NOTE = {MR:0130679. Zbl:0111.18606.},
ISSN = {0030-8730},
}
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
Abstract
BibTeX
@article {key0131265m,
AUTHOR = {Bing, R. H.},
TITLE = {A surface is tame if its complement
is 1-{ULC}},
JOURNAL = {Trans. Am. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical
Society},
VOLUME = {101},
NUMBER = {2},
MONTH = {November},
YEAR = {1961},
PAGES = {294--305},
DOI = {10.2307/1993375},
NOTE = {MR:0131265. Zbl:0109.15406.},
ISSN = {0002-9947},
}
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
Abstract
BibTeX
@article {key0146812m,
AUTHOR = {Bing, R. H.},
TITLE = {Each disk in \$E^3\$ is pierced by a tame
arc},
JOURNAL = {Am. J. Math.},
FJOURNAL = {American Journal of Mathematics},
VOLUME = {84},
NUMBER = {4},
MONTH = {October},
YEAR = {1962},
PAGES = {591--599},
DOI = {10.2307/2372865},
NOTE = {MR:0146812. Zbl:0178.27202.},
ISSN = {0002-9327},
}
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
Abstract
BibTeX
@article {key0146811m,
AUTHOR = {Bing, R. H.},
TITLE = {Each disk in \$E^3\$ contains a tame arc},
JOURNAL = {Am. J. Math.},
FJOURNAL = {American Journal of Mathematics},
VOLUME = {84},
NUMBER = {4},
MONTH = {October},
YEAR = {1962},
PAGES = {583--590},
DOI = {10.2307/2372864},
NOTE = {MR:0146811. Zbl:0178.27201.},
ISSN = {0002-9327},
}
R. H. Bing :
“Approximating surfaces from the side ,”
Ann. Math. (2)
77 : 1
(January 1963 ),
pp. 145–192 .
MR
0150744
Zbl
0115.40603
article
Abstract
BibTeX
@article {key0150744m,
AUTHOR = {Bing, R. H.},
TITLE = {Approximating surfaces from the side},
JOURNAL = {Ann. Math. (2)},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {77},
NUMBER = {1},
MONTH = {January},
YEAR = {1963},
PAGES = {145--192},
DOI = {10.2307/1970203},
NOTE = {MR:0150744. Zbl:0115.40603.},
ISSN = {0003-486X},
}
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
Abstract
People
BibTeX
In this paper we give a complete and self-contained proof that real Hilbert space, \( l_2 \) , is homeomorphic to the countable infinite product of lines, \( s \) ; symbolically \( l_2\sim s \) .
@article {key0230284m,
AUTHOR = {Anderson, R. D. and Bing, R. H.},
TITLE = {A complete elementary proof that {H}ilbert
space is homeomorphic to the countable
infinite product of lines},
JOURNAL = {Bull. Am. Math. Soc.},
FJOURNAL = {Bulletin of the American Mathematical
Society},
VOLUME = {74},
NUMBER = {5},
YEAR = {1968},
PAGES = {771--792},
DOI = {10.1090/S0002-9904-1968-12044-0},
NOTE = {MR:0230284. Zbl:0189.12402.},
ISSN = {0002-9904},
}
R. H. Bing :
“The elusive fixed point property ,”
Am. Math. Mon.
76 : 2
(February 1969 ),
pp. 119–132 .
MR
0236908
Zbl
0174.25902
article
Abstract
BibTeX
A set \( X \) has the fixed point property if each map \( f:X\to X \) leaves some point fixed — that is, there is a point \( x\in X \) such that \( f(x)=x \) . Satisfactory necessary and sufficient condidtions have not been found for determining whether or not a set has the fixed point property. The sufficient conditions that have been found are too restrictive to be necessary. On the other hand, many examples have been shown to have the fixed point property — sometimes with the method of proof tailored to the example. The paper is intended primarily as an expository article to bring together some of the interesting results about fixed points.
@article {key0236908m,
AUTHOR = {Bing, R. H.},
TITLE = {The elusive fixed point property},
JOURNAL = {Am. Math. Mon.},
FJOURNAL = {The American Mathematical Monthly},
VOLUME = {76},
NUMBER = {2},
MONTH = {February},
YEAR = {1969},
PAGES = {119--132},
DOI = {10.2307/2317258},
NOTE = {MR:0236908. Zbl:0174.25902.},
ISSN = {0002-9890},
}
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
Abstract
People
BibTeX
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 \( \operatorname{Bd}I^3 - \operatorname{Int}I^2 \) .
@incollection {key0275379m,
AUTHOR = {Bing, R. H.},
TITLE = {The monotone mapping problem},
BOOKTITLE = {Topology of manifolds},
EDITOR = {Cantrell, J. C. and Edwards, C. H.},
SERIES = {Markham Mathematics Series},
PUBLISHER = {Markham},
ADDRESS = {Chicago},
YEAR = {1970},
PAGES = {99--115},
NOTE = {(University of Georgia, Athens, GA,
11--22 August 1969). MR:0275379. Zbl:0283.57004.},
}
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
BibTeX
@book {key728227m,
AUTHOR = {Bing, R. H.},
TITLE = {The geometric topology of 3-manifolds},
SERIES = {AMS Colloquium Publications},
NUMBER = {40},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {1983},
PAGES = {x+238},
NOTE = {MR:728227. Zbl:0535.57001.},
ISSN = {0065-9258},
ISBN = {0821810405, 9780821810408},
}