What is the fundamental group $\pi (S^2, (0,0,1)) $ of the unit sphere with base point $ (0,0,1)$?

Question

I'm trying to find the fundamental group of the unit sphere with base point $(0,0,1)$. Does it matter that my base point is $(0,0,1)$ specifically?

Answer 1

In a path connected space $X$, $\pi_1(X,x)\cong\pi_1(X,x')$ for all $x$, $x'\in X$.

The space $S^n$ is well-known to be simply connected for $n\ge2$.