1. ## Topology: Equivalent Spaces

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 $\mathbb{R}^2$:

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

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

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

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

2. 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 $\mathbb{R}^2$:

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

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

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

I would appreciate your ideas on how to prove that S1 and S2 are homeomorphic. Thank you.
Try working piecewise (show the individual elements of the "small unions" are homeomorphic) and then try to apply the gluing lemma to produce a big homeomorphism. Intuitively they look like they're homeomorphic, I agree. But you still need to write out the homeomorphism.