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Most of the examples of fundamental groups have been "discrete" in nature. I'm wondering if there is an example which is not like that. A possible preliminary question could be if there is a space whose fundamental group is the rationals.

Edit: This has been marked as a duplicate; it is with respect to my subquestion about rationals however the linked answer does not say anything about topological spaces whose fundamental groups are the real numbers.

edenstar
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    Every group is the fundamental group of some space. – Mariano Suárez-Álvarez Dec 01 '17 at 00:00
  • The real or rational numbers just viewed as a group don't have any intrinsic continuous structure. – Rob Arthan Dec 01 '17 at 01:06
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    Your edit is not quite correct: read the accepted answer there. – Mariano Suárez-Álvarez Dec 01 '17 at 03:56
  • right, it says that for every group G there is a space whose fundamental group is G. However I was hoping to get an example in the case of $\mathbb{R}$. I suppose you could use the presentation space in one of the answers but I don't see exactly how that would work. – edenstar Dec 01 '17 at 04:04
  • Every group $G$ has a canonical presentation whose generators consist of every element $g \in G$ and whose relations consist of the entire multiplication table $g \cdot h = gh$ of $G$. So you can always take the presentation complex associated to this presentation. (Carrying this idea further leads to a construction of what is called the classifying space $BG$ of $G$.) – Qiaochu Yuan Dec 01 '17 at 06:16

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