Reference articles on history, science, culture and more
Surprise meSurprise meSubscribe
Encyclopedia

Dogbone space

Quotient space in geometric topology

Wikipedia contributorsAdapted for Fathomly · View authors
1 min read
Source updated July 6, 2026
Image credit is listed at the end of this article.

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

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

Watch videos about Dogbone spaceExplainers and documentaries on YouTube (opens in a new tab)

Sources and credits

This article is adapted from the Wikipedia article “Dogbone space”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

Images, from Wikimedia Commons:

Fathomly is not affiliated with or endorsed by the Wikimedia Foundation. Spotted a problem? Tell us.

Continue exploring

Related topics

List of topologies

The following is a list of named topologies or topological spaces, many of which are counterexamples in topology and related branches of mathematics. This is not a list of properties that a topology or topological space might possess; for that, see List of general topology topics and Topological property.

Whitehead manifold

In mathematics, the Whitehead manifold is an open 3-manifold that is contractible, but not homeomorphic to R 3 . {\displaystyle \mathbb {R} ^{3}.} It was discovered by J. H. C. Whitehead while trying to prove the Poincaré conjecture, correcting an error in an earlier paper Whitehead (1934, theorem 3) where he incorrectly claimed that no such manifold exists.