Dogbone space

In geometric topology, the dogbone space, constructed by R. H. Bing (1957), is a quotient space of three-dimensional Euclidean space R³ such that all inverse images of points are points or tame arcs, yet it is not homeomorphic to R³. The name "dogbone space" refers to a fanciful resemblance between some of the diagrams of genus 2 surfaces in R.H. Bing's paper and a dog bone. Bing (1959) showed that the product of the dogbone space with R¹ is homeomorphic to R⁴.

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