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So I'm working on a proof for $g(\partial U) = \partial g(U)$ that uses the following assumptions:

$W \subset \mathbb{R}^n$ open.
$g: W \rightarrow g(W)$ diffeomorphism.
$U \subset W$ bounded for which $\bar{U} \subset W$.

Additionally the proof uses the boundary $\partial U$.

What I'm confused about is why is "$U \subset W$ bounded" as an assumption here?

  • 1
    What is the goal of the proof? – 2017-02-16
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    @tommyxu3 $g(\partial U) = \partial g(U)$ – 2017-02-16

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Let be $W =$ an infinite horizontal band, $g =$ squeeze horizontally $W$ to a square. What happens when $U =$ smaller infinite horizontal band?