Let be the disk of radius centered at with basepoint . Let so that is the usual infinite earring. Let be a closed disk contained in the open set .
Embed in the xy-plane of . If is the center of be the cone in over with vertex . The (metric) Harmonic Archipelago is
More simply, view as a subspace of and push up a hill of height between each pair of consecutive circles and .
Topological Properties of the construction in 3-space: 2-dimensional, path-connected, locally path-connected, metric space. Not compact.
Relative CW construction: Let be a homeomorphism that generates . For each , attach a 2-cell to by the attaching loop .
Topological Properties of the relative CW construction: 2-dimensional, path-connected, locally path-connected, paracompact Hausdorff space. Not compact. Not first countable at . Homotopy theory is easier to do in this construction since every compact set may intersect at most finitely many of the attached 2-cells . The metric and relative CW constructions are homotopy equivalent.
Fundamental Group: Using the relative CW construction, it is clear that is isomorphic to the quotient where is the normal closure of in . One of the consequences of this is that in .
Alternatively, let be the map, which is determined by , i.e. shifting the -th circle to the -st circle and the identity elsewhere. Then there is a canonical isomorphim to the direct limit
Fundamental Group Properties: Uncountable, torsion free, locally free. Not residually free.
Higher homotopy groups: for , i.e. is aspherical.
Cech homotopy groups: for all .
Cech homology groups: for all .
The Cech homotopy and homology groups are trivial even though nerves of carefully chosen open covers of are each homotopy equivalent to an infinite wedge of 2-spheres.
Cech cohomology groups:
Wild Set/Homotopy Type: The wild set is a single point.
- Semi-locally simply connected: No.
- Traditional Universal Covering Space: No
- Generalized Universal Covering Space: No
- Homotopically Hausdorff: No
- Strongly (freely) Homotopically Hausdorff: No
- Homotopically Path-Hausdorff: No
- –: No
- -shape injective: No
 W. Bogley, A. Sieradski, Universal path spaces, Unpublished manuscript.
 G. Conner, W. Hojka, M. Meilstrup, Archipelago Groups. Proceedings of the American Mathematical Society. 143 no. 11 (2015) 4973–4988.
 P. Fabel, The fundamental group of the harmonic archipelago. Unpublished manuscript. Arxiv.
 U.H. Karimov, D. Repovš, On the homology of the harmonic archipelago, Cent. Eur. J. Math. 10 (2012) 863–872.