I am studying Elementary Differential Geometry written by Barrett O'Neill.
In page 188, Chapter 4.7, there is an explanation why 2-sphere is simply connected.
The following is from the text :
The 2-sphere $\Sigma$ is simply connected. Consider the following scheme of proof. Let $\alpha$ be a loop in $\Sigma$ at, say, the north pole of $\Sigma$. Pick a point q not on $\alpha$. For simplicity, suppose q is the south pole. Now let x be the homotopy under which each point of $\alpha$ moves due north along a great circle, reaching p in unit time. This x is a homotopy of $\alpha$ to a constant, as required.
But there is a difficulty here: finding the point q. In our usual case, where $\alpha$ is differentiable, techniques from advanced calculus will show that there is always a point q not on $\alpha$. However, if $\alpha$ is merely continuous, it may actually fill the entire sphere. In this case, topological methods can be used to deform $\alpha$ slightly, making it no longer space-filling: then the scheme above is valid.
My questions are:
Why why why why does he pick q ??? for what ???
I cannot even grasp What he wants to say in the 2nd paragraph. Can you explain why the 2nd paragraph is needed?