Zeroth homotopy group: what exactly is it? What are the elements in the zeroth homotopy group? Also, why does $\pi_0(X)=0$ imply that the space is path-connected?
Thanks for the help. I find that zeroth homotopy groups are rarely discussed in literature, hence having some trouble understanding it. I do understand that the elements in $\pi_1(X)$ are loops (homotopy classes of loops), trying to see the relation to $\pi_0$.

This is the definition.
So $\pi_0$ is the homotopy classes of maps from two points ($S^0$) to $X$, where the first point is mapped to the base point.
Clearly only the path connected component matters for the second point (since a path connecting the two points defines a homotopy between two such maps).
$\pi_0$ being trivial implies that there is a path between any point and the base point, i.e. $X$ is path-connected.
