The double earring consists of two copies of the usual infinite earring adjoined by an arc. Specifically, one many construct it as:
where and are the circle of radius centered at and respectively.
Topological Properties: Planar, 1-dimensional, path-connected, locally path-connected, compact metric space.
Fundamental Group: Using the van Kampen Theorem, one can show that is isomorphic to the free product of the earring group with itself, i.e. . It is an interesting and non-obvious fact that this fundamental group is not isomorphic to itself.
Fundamental Group Properties: Uncountable, residually free, torsion free, locally free, locally finite.
Higher homotopy groups: for , i.e. this space is aspherical.
is isomorphic (but not naturally) to .
Cech Homotopy groups:
Cech Homology groups:
We have a summand of in for both copies of in . However, .
Wild Set/Homotopy Type: The 1-wild set is the two-point discrete set . Since the homotopy type of the 1-wild set is a homotopy invariant, one can see from this that is not homotopy equivalent to . In fact, any map which induces an injection on must map both wild points of to the wild-point of and then cannot induce a surjection on using standard alternating infinite product arguments.
- Semi-locally simply connected: No, not at nor .
- Traditional Universal Covering Space: No
- Generalized Universal Covering Space: Yes
- Homotopically Hausdorff: Yes
- Strongly (freely) homotopically Hausdorff: Yes
- Homotopically Path-Hausdorff: Yes
- –: Yes
- -shape injective: Yes
 J.W. Cannon, G.R. Conner, On the fundamental groups of one-dimensional spaces, Topology Appl. 153 (2006) 2648–2672.