Is there a space with fundamental group isomorphic to $\mathbb{R}$?

Question

I am studying basic algebraic topology and all examples of fundamental groups $\mathbb{Z}, \mathbb{Z^n \times \mathbb{Z}} / n \mathbb{Z} $, $\mathbb{Z}*\mathbb{Z}$ etc are somewhat discrete. I don't have a rigorous way of stating this right now, but basically they are countable. Moreover, neither is isomorphic to $\mathbb{R}$.

Now, apart from rigorous treatment of why a circle has integers as its FG, there is a moral to it : you can't loop/unloop in fractions or in irrationals units; you either unloop or you didn't.

My question is, is there a space where you can have a FG that is not countable, or if possible is $\mathbb{R}$, so that you can loop or unloop continuously.

Term loop/unloop are used unrigorously.

Answer 1

"Discrete" and "Countable" are two different things. There are certainly spaces with uncountable fundamental groups; you can achieve any cardinality you like. As for discreteness, by definition the fundamental group does not come endowed with any topology, so it is (again, by definition) discrete.

In fact, every group is the fundamental group of some space. So, yes, that includes $\mathbb{Q}$ and $\mathbb{R}$ (as discrete groups). See, for example, the Classifying space, which has the added feature that its higher homotopy groups are trivial.

Answer 2

One way to topologize $\pi_1(X, x_0)$ in a canonical way is to define it as $\pi_0$ of the loop space $\Omega(X, x_0)$. (Note that $\pi_0$ carries a canonical topology since $\pi_0(Y)$ is a set-theoretic quotient of $Y$.)

With this description, one thing you get immediately is that $\pi_1(X)$ has to be totally disconnected, so e.g. $\mathbb{R}$ won't show up. However, it doesn't need to be discrete -- I believe the Hawaiian earring is a counterexample.