Is homeomorphic image of convex set is convex?

Question

Is homeomorphic image of a convex set is convex?

Answer 1

Here's a counterexample:

The set $A = \{(x, 0) : x \in \mathbb R\}$ is convex (it's just the $x$-axis of the plane). But the set $B = \{(x, \sin x) : x \in \mathbb R$} is not convex. The map $f : A \to B$ defined by $f(x, 0) = (x, \sin x)$ is a homeomorphism of these sets.