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Is homeomorphic image of a convex set is convex?

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    No. Take a disk and just stretch and bend it. – 2017-02-12

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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.