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 $\pi_0$.

Just a slight rephrase: you can consider $\pi_0(X)$ as the quotient set of the set of all points in $X$ where you mod out by the equivalence relation that identifies two points if there is a path between them.
