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), $A = \lbrace a_n \rbrace$, the union of circles of radius $a_n$ centered at $(0, a_n )$.  Let the union inherit its topological structure from $\mathbb{R}^2$.  Are all of these spaces homeomorphic?

If $a_n$ is a monotone decreasing sequence converging to $0$, is its Hawaiian Earring homeomorphic to that of the sequence $\lbrace 1/n \rbrace$?


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