The Hawaian earring is the one-point compactification of a countable union of open intervals (with the coproduct or disjoint sum topology). 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](http://www.ams.org/mathscinet-getitem?mr=MR1775709)) If I recall correctly, they prove my claim there.