I'm studying weighed projective spaces and I found this reference http://arxiv.org/pdf/math/0510331v1.pdf* where it describes its orbifold charts (starts at page 53 of the PDF). My doubt is very simple: an orbifold chart is defined as a triple $(\tilde{U}, G, \phi)$, with $\tilde{U}$ an open connected set of $\mathbb{R}^n$ or $\mathbb{C}^n$. But the author claims the triples $(\tilde{U}_i, \mu_{a_i}, \phi_i)$ are orbifold charts for the weighed projective space, with $\tilde{U}_i=\{y \in \mathbb{C}^{n+1}-\{0\}; y_i=1\}$, which is not open.
Could the author be identifying $\tilde{U}_i$ with $\mathbb{C}^n-\{0\}$, so that it is seen as a subset of $\mathbb{C}^n$?
*The text is in French. There is a slightly less-detailed version in English here http://arxiv.org/pdf/math/0610965.pdf