Path-connected does not imply convex

Question

I'm trying to prove that A subset of the Complex numbers which is path-connected does not imply that it is convex.

Any help?

Answer 1

Hint: (1) Is the unit circle (not disk) path connected? (2) Is a deleted neighbourhood of a point convex?

Answer 2

I interpret "does not imply" as "look for a counterexample".

Convex means that the straight line between any two points in a set in still in that set.

Path connected means that you can find a path in the set between any two points in that set, which does not have to be a straight line.

So you could look at $\mathbb{C}\backslash \{0\}$ for example and try out the definitions on the points $-1$ and $1$.

Answer 3

Consider any path that isn't a straight line or line segment. This set is path connected but not convex.

Answer 4

Let $A,B$ and $C$ be the points inside the region ${A}$.