Let be the circle of radius centered at with basepoint . Let be the odd circles of the earring space and be the even circles so that the union is the usual earring space and . Embed in the xy-plane. Let be the cone over the odd circles with vertex and be the cone over the even circles with vertex . The Griffiths Twin Cone is .
Generic Construction: Since and are homeomorphic copies of , one can construct by letting be the cone over the earring with basepoint , the image of in the quotient. Then is the wedge .
Topological Properties: 2-dimensional, path-connected, locally path-connected, compact metric space. Embeds in .
Fundamental Group: The inclusions and are sections and so induce injections on . Let and be the images of these homomorphisms. By the van Kampen Theorem, is isomorphic to where is the normal closure of .
Alternatively, let be the retractions that (respectively) collapse and to and are the identity elsewhere. Then is isomorphic to the coequalizer of the induced homomorphisms .
Fundamental Group Properties: Uncountable, torsion free, locally free. Not residually free.
Higher homotopy groups: for , i.e. is aspherical.
Note: the isomorphism in dimension 1 is not constructive.
Cech Homotopy groups: for all . Moreover, is shape equivalent to a point.
Cech Homology groups: for all .
Wild Set/Homotopy Type: The wild set is a single point.
- Semi-locally simply connected: No.
- Traditional Universal Covering Space: No
- Generalized Universal Covering Space: No
- Homotopically Hausdorff: No
- Strongly (freely) Homotopically Hausdorff: No
- Homotopically Path-Hausdorff: No
- –: No
- -shape injective: No
 H.B. Griffiths, The fundamental group of two spaces with a common point, Quart. J. Math. Oxford (2) 5 (1954) 175-190.
 H. Fischer, K. Eda, Cotorsion-free groups from a topological viewpoint, Topology Appl. 214 (2016) 21-34.
 K. Eda, A locally simply connected space and fundamental groups of one point unions of cones, Proceedings of the American Mathematical Society. 116 no. 1 (1992) 239-249.