Smooth effective orbifold

A smooth effective orbifold of dimension $n$ is a Hausdorff, second-countable space $X$ equipped with an equivalence class of compatible local quotient atlases. A chart consists of a connected open set $\widetilde U\subseteq\mathbb R^n$, a finite group $G$ acting smoothly and effectively on it, and a continuous map $\phi:\widetilde U\to X$ inducing a homeomorphism $\widetilde U/G\cong U$ onto an open subset of $X$.

The chart images cover $X$. At each point of two overlapping images there must be a third chart mapping into both: a chart embedding consists of an injective group homomorphism and an equivariant smooth open embedding of the chart domains that commutes with the maps to $X$. Atlases are equivalent when they admit a common refinement. These compatibility conditions, together with the local finite-quotient charts, are the orbifold structure.