A topological space is called arcwise connected if, for any points $x,y\in X$, there exists a continuous map $f: [0,1]\rightarrow X$ such that $f(0)=x$ and $f(1)=y$. Although it is intuitively understandable but how does such a map mathematically look like for $S^2$?
According to this definition is there way to show that $SU(2)$ is connected but $O(3)$ is not? As I continuously change the group parameters (up to their allowed ranges) can I show in the first case that I can reach all points on the $SU(2)$ manifold and in case of $O(3)$ I cannot exhaust all points? Only this can prove the nature of connectedness, in this definition, Right?