A pair of integers $(n,j)$ is dyadic unital if the dyadic rational $\frac{2j-1}{2^n}$ lies in $(0,1)$. For each dyadic unital pair $(n,j)$ consider the upper semicircle:

$\mathbb{D}(n,j)=\left\{(x,y)\in \mathbb{R}^2\mid\left(x-\frac{2j-1}{2^n}\right)^2+y^2=\left(\frac{1}{2^n}\right)^2\text{, }x\geq 0\right\}$.

Let $B=[0,1]\times \{0\}$ be the base arc. The dyadic arc-space is the union

$\mathbb{D}=B\cup \bigcup_{(n,j)} \mathbb{D}(n,j)$

over all dyadic unital pairs with the subspace topology inherited from $\mathbb{R}^2$ and with basepoint $d_0=(0,0)$.

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

Utility: Used in [2] to characterize the existence of generalized universal coverings (as defined in [3] and extended in [1]).

Fundamental Group: $\pi_1(\mathbb{D})$ embeds as the subgroup of the natural inverse limit $\varprojlim_{n}F_{2^{n}-1}$ of finitely generated free groups. This group may also be described in terms of a reduced path calculus.

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

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

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

Cech Homotopy groups: $\check{\pi}_n(\mathbb{D})=\begin{cases} \varprojlim_{n}F_{2^{n}-1}, & n=1 \\ 0, & n\neq 1 \end{cases}$

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

Wild Set/Homotopy Type: The wild set is uncountable, namely, the arc $\mathbf{w}(\mathbb{D})=B$. Since $\mathbb{D}$ is a one-dimensional Peano continuum any other 1-dim. Peano continuum with fundamental group isomorphic to $\pi_1(\mathbb{D})$ must be homotopy equivalent to $\mathbb{D}$.

The dyadic arc space is homotopy equivalent to the “triangle continuum” seen on the right, which appears in [3] and [4].

Other Properties:

• Semi-locally simply connected: No, not at any point of $B$
• 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

Other Note: If you collapse the base arc to a point, you get a space homeomorphic to the infinite earring.

References:

[1] J. Brazas, Generalized covering space theories, Theory and Appl. of Categories 30 (2015) 1132-1162

[2] Brazas, Fischer, Test map characterizations of local properties of fundamental groups. Journal of Topology and Analysis. 12 (2020) 37-85. arXiv.

[3] H. Fischer, A. Zastrow, Generalized universal covering spaces and the shape group, Fundamenta Mathematicae 197 (2007) 167–196.

[4] Z. Virk, A. Zastrow, A homotopically Hausdorff space which does not admit a generalized universal covering, Topology Appl. 160 (2013) 656–666.