Given $[0,1] \in \mathbb{R}$ and $E$ a metric space, $f:[0,1] \to E$. How to prove $E$ is connected? Is it to prove that $f([0,1]) =E$?
I come up with this idea because my text uses a continuous function on $[0,1]$ to prove all ball in Euclidean space, and Euclidean space itself, are connected after introducing intermediate value theorem. I also read the Wikipedia, but there are some concepts that I don't have a good understanding yet. Such as path, curve, path connectedness, arc connectedness.
Also, what're the differences between path connectedness and arc connectedness?
