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  to characterize the existence of generalized universal coverings (as defined in  and extended in ).
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.
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  and .
- 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.
 J. Brazas, Generalized covering space theories, Theory and Appl. of Categories 30 (2015) 1132-1162
 Brazas, Fischer, Test map characterizations of local properties of fundamental groups. Journal of Topology and Analysis. 12 (2020) 37-85. arXiv.
 H. Fischer, A. Zastrow, Generalized universal covering spaces and the shape group, Fundamenta Mathematicae 197 (2007) 167–196.
 Z. Virk, A. Zastrow, A homotopically Hausdorff space which does not admit a generalized universal covering, Topology Appl. 160 (2013) 656–666.