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snesnerd

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## Homework Statement

I recently read and proved that the real line is not homeomorphic to the circle. It was very interesting and made sense to me. I started pondering to myself and thought of another interesting problem. What if I considered the circle [itex]S^1[/itex] and [itex]D[/itex] where [itex]D[/itex] is the open disk. The question I asked myself is the following: Is [itex]S^1 \approx D [/itex] where [itex]\approx[/itex] signifies a homeomorphism. I thought about it and think that they are not.

## Homework Equations

## The Attempt at a Solution

My idea goes as follows. Do a proof by contradiction and use the idea of what it means to be connected and not connected. Suppose they were homeomorphic. Clearly if I remove a point from [itex]S^1[/itex] then it is still connected. Same goes for [itex]D[/itex]. It does not seem helpful at first, but what if I remove another point from each? I think I get one of them to be connected and one of them to be not connected which is a contradiction. Of course this is my idea. I would like to write it out nicely, so if anyone could help me out it would be greatly appreciated! I wasn't sure if this was the way to go.

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