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I stumbled upon this excerpt as I was reading Graph Theory by Reinhard Diestel:

A polygon is a subset of $\mathbb{R}^2$ which is the union of finitely many straight line segments and is homeomorphic to the unit circle $S^1$, the set of points in $\mathbb{R}^2$ at distance 1 from the origin.

So based on this, how could any polygon be homeomorphic to $S^1$ even though both sets are of different cardinality?

Pardon me if the question is too basic; I'm totally new to topology and I probably am overlooking a detail.

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    Why do the two sets have different cardinality ?2012-12-05
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    Why do you think the sets have different cardinality? A polygon is the set of points defined by the union of the sets that are defined by line segments, not the set of line segments.2012-12-05
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    I'm _assuming_ so, I honestly don't know.2012-12-05
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    OK, why are you assuming it?2012-12-05
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    In that case, the first thing you should do is find a bijection between a line segment like $[0,1)$ and $S^1$.2012-12-05
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    I just realized what my mistake was.. Thanks a lot :)2012-12-05

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I'll give a conceptual answer to your question. Two curves are homeomorphic if the first can be continuously deformed into the second. Intuitively, if you had a circular loop of wire, you could hammer and bend it into the shape of any polygon (without breaking the wire or adding any new connections).

The cardinality of $S^1$ is the same as the cardinality of any line segment, which in turn has the same cardinality as any finite union of line segments. Cardinality of a set is different from "length" or "measure". For example, though $[1,2]$ is a proper subset of $[0,3]$, the two sets have the same cardinality.