Let $ B \subseteq \mathbb{R}^n$ be a product of closed bounded intervals in $ \mathbb{R}$ . Fix $ N>0$ . Suppose I want to cover $ B$ with $ N$ open sets, $ U_1, \ldots, U_N$ , and get a smooth partition of unity $ \rho_1, \ldots, \rho_N$ with respect to these sets. I was wondering is it possible to do this in a way that the derivatives of the $ \rho_j$ ‘s are bounded independent of the choice of the $ U_j$ ‘s?

I was wondering maybe I am asking for too much here and that this is not possible, or maybe it’s possible? I have no idea… I would appreciate any comments or suggestions. Thank you.

PS I would like to change the question slightly. I would like to assume that each $ U_j$ is not too small in that each $ U_j$ contains an open set of the form $ (x_1 – \varepsilon, x_1+ \varepsilon) \times \cdots \times (x_n – \varepsilon, x_n+ \varepsilon)$ for some $ \varepsilon > 0$ .