Originally Posted by

**huram2215** I have been given several spaces and asked to find which ones are equivalent. I believe that the following two spaces are homeomorphic. Each are harmonic sets (easy to represent in a drawing if you take a moment) and are subpaces of $\displaystyle \mathbb{R}^2$:

1) $\displaystyle \ S1 = ([0,1] \times \{0\}) \cup (\{0\} \times [0,1]) \cup ( \bigcup_{n=1}^{\infty}{(\{ 1/n \} \times [0,1]}}))$

2) $\displaystyle S2 = (\{0 \} \times [-1,1]) \cup ([0,1] \times \{ 1 \}) \cup (\bigcup_{n=1}^{\infty}{(\{1/n \} \times [0,1]})) $

$\displaystyle \cup (\bigcup_{n = 1}^{\infty}{(\{-1/n\} \times [-1,0])}) \cup ([-1,0] \times \{ -1\})$

I would appreciate your ideas on how to prove that S1 and S2 are homeomorphic. Thank you.