**Other Names:** infinite earring, 1-dimensional earring, shrinking wedge of circles, shrinking bouquet of circles, clamshell. This space has often been referred to as the “Hawaiian earring” in the literature; however, there is substantial movement by experts to stop using this term.

**Topological Properties:** Planar, 1-dimensional, path-connected, locally path-connected, compact metric space.

**Other constructions:**

- Reduced suspension of -dimensional earring, i.e. the convergent sequence space
- infinite wedge viewed as subspace of infinite dimensional torus with the product topology.
- one-point compactification of a countable disjoint union of open arcs.

**Universal Property:** given any sequence of loops based at converging (in the compact-open topology) to the constant loop at , there exists a unique map such that where is the canonical counterclockwise loop traversing the n-th circle. This universal property is a special case of the loop space-suspension adjuction.

**Fundamental Group:** embeds as the subgroup of the natural inverse limit of finitely generated free groups consisting of locally eventually constant elements. This group is is sometimes called the free--product and denoted . See this post and others for more details.

**Fundamental Group Properties:** Uncountable, Residually free, torsion free, locally free, locally finite.

**Higher homotopy groups:** for , i.e. is aspherical.

**Homology groups:**

**Cech homotopy groups:**

**Cech homology groups:**

**Cech cohomology groups:**

**Wild Set/Homotopy Type:** The wild set is the single-point set . The infinite earring represents the unique homotopy type of 1-dimensional Peano continua with a single wild point.

**Other Properties:**

**Semi-locally simply connected:**No, not at .**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:**

There are a couple hundred papers involving the 1-dimensional earring. Here are a couple key ones:

[1] J.W. Cannon, G.R. Conner, *The combinatorial structure of the Hawaiian earring group*, Topol. Appl. 106 (2000) 225–271.

[2] K. Eda, *Free σ-products and noncommutatively slender groups*, J. Algebra 148 (1992) 243–263.

[3] K. Eda, K. Kawamura, *The singular homology of the Hawaiian Earring*, J. Lond. Math. Soc. (2) 62 (2000) 305–310.

[4] J.W. Morgan, I. Morrison, *A van Kampen theorem for weak joins*, Proc. Lond. Math. Soc. (3) 53 (1986) 562–576.