Earring Space

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Wikipedia
the Infinite Earring - Scientific American (译文)
Earring Space | Wild Topology

The infinite earring$$\mathbb{E}=\bigcup_{n\in\mathbb{N}}\left\{(x,y)\in\mathbb{R}^2\mid (x-1/n)^2+y^2=1/n^2\right\}$$

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 $0$-dimensional earring, i.e. the convergent sequence space $\{0,\dots,1/3,1/2,1\}$
• infinite wedge $\bigvee_{\mathbb{N}}S^1$ viewed as subspace of infinite dimensional torus $\prod_{\mathbb{N}}S^1$ with the product topology.
• one-point compactification of a countable disjoint union of open arcs.

Universal Property: given any sequence $\{\alpha_n\}_{n\in\mathbb{N}}\to c_{x_0}$ of loops based at $x_0$ converging (in the compact-open topology) to the constant loop at $x_0$, there exists a unique map $f:\mathbb{E}\to X$ such that $f\circ\ell_n=\alpha_n$ where $\ell_n:S^1\to\mathbb{E}$ 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: $\pi_1(\mathbb{E})$ embeds as the subgroup of the natural inverse limit $\varprojlim_{n}F_n$ of finitely generated free groups consisting of locally eventually constant elements. This group is $\pi_1(\mathbb{E})$ is sometimes called the free-$\sigma$-product and denoted $\#_{\mathbb{N}}\mathbb{Z}$. See this post and others for more details.

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

Higher homotopy groups: $\pi_n(\mathbb{E})=0$ for $n\geq 2$, i.e. $\mathbb{E}$ is aspherical.

Homology groups: $\widetilde{H}_n(\mathbb{E})=\begin{cases} \prod_{\mathbb{N}}\mathbb{Z}\oplus\prod_{\mathbb{N}}\mathbb{Z}/\bigoplus_{\mathbb{N}}\mathbb{Z}, & n=1 \\ 0, & n \neq 1   \end{cases}$

Cech homotopy groups: $\check{\pi}_n(\mathbb{E})=\begin{cases}  \varprojlim_{n}F_{n}, & n=1 \\ 0, & n\neq 1   \end{cases}$

Cech homology groups: $\check{H}_n(\mathbb{E})=\begin{cases} \mathbb{Z}, & n=0 \\ \prod_{\mathbb{N}}\mathbb{Z}, & n=1 \\ 0, & n \geq 2   \end{cases}$

Cech cohomology groups: $\check{H}_n(\mathbb{E})=\begin{cases} \mathbb{Z}, & n=0 \\ \bigoplus_{\mathbb{N}}\mathbb{Z}, & n=1 \\ 0, & n \geq 2   \end{cases}$

Wild Set/Homotopy Type: The wild set is the single-point set $\mathbf{w}(\mathbb{E})=\{(0,0)\}$. 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 $(0,0)$.
• Traditional Universal Covering Space: No
• Generalized Universal Covering Space: Yes
• Homotopically Hausdorff: Yes
• Strongly (freely) homotopically Hausdorff: Yes
• Homotopically Path-Hausdorff: Yes
• $1$–$UV_0$: Yes
• $\pi_1$-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.