I have the following definition:
Given a topological space X, and a basepoint $x_0 \in X$, we define:
$\pi_1(X,x_0)$ := {homotopic equivalence classes of loops in X with basepoint $x_0$}. This is called the fundamental group of X with basepoint $x_0$.
I need to find $\pi_1(\mathbb{R}\backslash\{0\},1)$.
I think $\pi_1(\mathbb{R}\backslash\{0\},1)$ = $\pi_1(\mathbb{R}_+,1)$, since the basepoint lies in $\mathbb{R}_+$ and loops have to be continuous functions, meaning that they can't jump over $\{0\}$.
I'm stuck here; isn't every pair of loops in $\mathbb{R}_+$ homotopic? How would I write down $\pi_1(\mathbb{R}\backslash\{0\},1)$ in that case?
Q: What is $\pi_1(\mathbb{R}\backslash\{0\},1)$ = $\pi_1(\mathbb{R}_+,1)$?
Indeed we are left with computing $\pi_1(\Bbb R^+, 1)$. Consider a loop $\gamma(t) \subset \Bbb R^+$, the homotopy $H(t,s) = (1-t)\gamma(s) + t$. $H(0,s) =\gamma(s)$ and $H(1,s) = 1$. This shows that any path $\gamma$ in $\Bbb R^+$ is homotopic to the constant path $1$, so $\pi_1(\Bbb R^+,1) = 0$.
Notice that such homotopy always work where our space is convex (more precisely star-shaped) : this shows that any such space have a trivial fundamental group, for example $\Bbb R^n$ or $D^n$.