The *double earring* consists of two copies of the usual earring space adjoined by an arc. Specifically, one many construct it as:

where and are the circle of radius centered at and respectively.

The double earring

**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.

**Homology groups:**

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.

**Other Properties:**

**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

**References:**

[1] J.W. Cannon, G.R. Conner, *On the fundamental groups of one-dimensional spaces*, Topology Appl. 153 (2006) 2648–2672.

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