Two neighborhoods of $0$ in the plane and the upper half plane (resp.) cannot be diffeomorphic.

Question

Let $\mathbb{R^2}$ be equipped with the standard topology, and let $\mathbb{H^2}$ be the upper half plane (containing the x-axis), equipped with the subspace topology. Let $U$ be an open neighborhood of $0$ in $\mathbb{R^2}$, $V$ an open neighborhood of $0$ in $\mathbb{H^2}$. How can I show that $U$ and $V$ are not diffeomorphic? My guess is that I should proceed by contradiction, and suppose that there is a diffeomorphism from U onto V. I feel like there will be an issue with the pre-image of a semi open ball centered at $0$, which should be open in $\mathbb{R^2}$. But I struggle to prove that properly.