The Hawaiian Earring 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_n}, the union of circles of radius a_n centered at (0, a_n). Let the union inherit its topological structure from R². 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 {1/n}?

The Hawaiian Earring 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 $\lbrace 1/n \rbrace$?