Connected Set And Convex Set

Question

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".

Answer 1

There is a relation. Pictures will help your intuition.

Every convex set is connected, but not the other way. Think about a barbell.

In a convex set every pair of points is joined by a line segment entirely within the set. In a more general connected set the "joining" need not be a line segment. (I put "joining" in quotes because there's no need here to go into the distinction between "connected" and "path connected".)

In one dimension (on the real line) convex and connected are the same.

Answer 2

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.

Answer 3

Every the convex sets are connected. Examples of connected sets, but not convex, are a torus, the place between concetric spheres.