Dogbone space

In geometric topology, the dogbone space, constructed by R. H. Bing,^[1] is a quotient space of three-dimensional Euclidean space ${\displaystyle \mathbb {R} ^{3}}$ such that all inverse images of points are points or tame arcs, yet it is not homeomorphic to ${\displaystyle \mathbb {R} ^{3}}$. 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 ${\displaystyle \mathbb {R} ^{1}}$ is homeomorphic to ${\displaystyle \mathbb {R} ^{4}}$.^[2]

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