{{Short description|Quotient space in geometric topology}} thumb|right|The first stage of the dogbone space construction. In geometric topology, the '''dogbone space''', constructed by R. H. Bing,<ref>{{Cite journal |last=Bing |first=R. H. |date=May 1957 |title=A Decomposition of E 3 into Points and Tame Arcs Such That the Decomposition Space is Topologically Different from E 3 |url=https://www.jstor.org/stable/1970058?origin=crossref |journal=The Annals of Mathematics |volume=65 |issue=3 |pages=484 |doi=10.2307/1970058|url-access=subscription }}</ref> is a quotient space of three-dimensional Euclidean space <math>\R^3</math> such that all inverse images of points are points or tame arcs, yet it is not homeomorphic to <math>\R^3</math>. The name "dogbone space" refers to a fanciful resemblance between some of the diagrams of genus 2 surfaces in Bing's paper and a dog bone. Bing showed that the product of the dogbone space with <math>\R^1</math> is homeomorphic to <math>\R^4</math>.<ref>{{Cite journal |last=Bing |first=R. H. |date=November 1959 |title=The Cartesian Product of a Certain Nonmanifold and a Line is E 4 |url=https://www.jstor.org/stable/1970322?origin=crossref |journal=The Annals of Mathematics |volume=70 |issue=3 |pages=399 |doi=10.2307/1970322|url-access=subscription }}</ref>
Although the dogbone space is not a manifold, it is a generalized homological manifold and a homotopy manifold.
== See also ==
* List of topologies * Whitehead manifold, a contractible 3-manifold not homeomorphic to <math>\R^3</math>.
==References== * * *
Category:Geometric topology Category:Topological spaces <references />
== Sources ==
* {{Citation|author1-link=Robert Daverman | last1=Daverman | first1=Robert J. | title=Decompositions of manifolds | journal=Geom. Topol. Monogr. | url=https://www.ams.org/bookstore-getitem/item=chel-362.h | isbn=978-0-8218-4372-7 | mr=2341468 | year=2007| volume=9 | pages=7–15 | doi=10.1090/chel/362| arxiv=0903.3055}}