The Hawaian earring is the one-point compactification of a countable union of open intervals. This description is independent of the radii used to construct it.
A beautiful reference about this space is [Cannon, J. W.; Conner, G. R. The combinatorial structure of the Hawaiian earring group. Topology Appl. 106 (2000), no. 3, 225--271 MR1775709) If I recall correctly, they prove my claim there.
