Connected Set And Convex Set I am taking my first course in Topology.
Intuitively it seems that there is a relation between a connected set and convex set. Moreover, I am trying to differentiate between the two, as they both seem to mean that the set consists of "one piece".

There is some connection (no pun intended) between those concepts.
Recall that  convex set is one which, given two points $a$ and $b$, the "line segment" given by $f(t)=(1-t)a+tb$ is entirely contained in that set.
So, by definition, a convex set is path-connected (and thus, connected). But do note that the converse is not true: the sphere $S^2 \subset\Bbb{R^3}$ (the surface only, not the interior) is connected and path-connected, but isn't convex.
