Definition:

The group $\pi_1(X, x_0)$ is called the fundamental group of the space X based at $x_0$. This group can be non-abelian although we need to do some work to produce an example. Indeed we need to do some work to produce such an example for which $\pi_1(X, x_0)$ is non-trivial. All we shall do in the rest of this lecture is to show that it is trivial in case $X$ is a convex subset of $\mathbb{R}^n$. First we shall see what happens when the base point is changed.