The [Hawaiian Earring][1] is usually constructed as the union of circles of radius 1/n centered at (0,1/n): $\bigcup_1^\infty \left[   (0, \frac{1}{n}) + \frac{1}{n}S^1 \right]$.  However, nothing stops us from using the sequence of radii $1/n^2$ or any other sequence of numbers $a_n$.

I will call a Hawaiian Earring for a sequence (of distinct real numbers), A = {a<sub>n</sub>}, the union of circles of radius a<sub>n</sub> centered at (0, a<sub>n</sub>).  Let the union inherit its topological structure from R<sup>2</sup>.  Are all of these spaces homeomorphic?

If a<sub>n</sub> is a monotone decreasing sequence converging to 0, is its Hawaiian Earring homeomorphic to that of the sequence {1/n}?


  [1]: http://en.wikipedia.org/wiki/Hawaiian_earring