The fundamental group of a point is $1$

Question

Show that the fundamental group of the point space $p$ is given as $\pi(p, w_0)=1$ where $w_0$ is the base point

This is probably somewhat trivial, but I am looking for a proof. I am familiar with computing fundamental groups by triangulating simplicial complexes. In this case the triangulation is itself with no generators, so it makes sense that the fundamental group is trivial

I am looking for an alternative, concrete proof

Thanks!

Answer 1

There is only one possible map from $[0, 1]$ to $p$, so there cannot be any two paths that are different, much less not even homotopic. Hence the fundamental group cannot have more than one element, so it is the trivial group.