A pair of integers is *dyadic unital* if the dyadic rational lies in . For each dyadic unital pair consider the upper semicircle:

.

Let be the *base arc*. The **dyadic arc-space** is the union

over all dyadic unital pairs with the subspace topology inherited from and with basepoint .

**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:** embeds as the subgroup of the natural inverse limit 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:**

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

**Cech Homotopy groups:**

**Cech Homology groups:**

**Wild Set/Homotopy Type:** The wild set is uncountable, namely, the arc . Since is a one-dimensional Peano continuum any other 1-dim. Peano continuum with fundamental group isomorphic to must be homotopy equivalent to .

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**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

**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.