Dogbone space

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title: "Dogbone space" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["geometric-topology", "topological-spaces"] description: "Quotient space in geometric topology" topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Dogbone_space" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Quotient space in geometric topology ::

::figure[src="https://upload.wikimedia.org/wikipedia/commons/c/c4/Bing's_Dogbone.tiff" caption="The first stage of the dogbone space construction."] ::

In geometric topology, the dogbone space, constructed by R. H. Bing, is a quotient space of three-dimensional Euclidean space \R^3 such that all inverse images of points are points or tame arcs, yet it is not homeomorphic to \R^3. 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 \R^1 is homeomorphic to \R^4.

Although the dogbone space is not a manifold, it is a generalized homological manifold and a homotopy manifold.

References

Sources

References

1. Bing, R. H.. (May 1957). "A Decomposition of E 3 into Points and Tame Arcs Such That the Decomposition Space is Topologically Different from E 3". The Annals of Mathematics.
2. Bing, R. H.. (November 1959). "The Cartesian Product of a Certain Nonmanifold and a Line is E 4". The Annals of Mathematics.

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