Two-Dimensional Orbifolds

1 Definitions An orbifold is a Hausdorff space X0 along with an open cover {Ui} that is closed under finite intersection with the following properties. For each Ui there is an open set Ũi ⊆ R and a finite group Γi which acts on Ũi. In addition there is a homeomorphism φi : Ui ≈ Ũi/Γi and whenever Ui ⊂ Uj there is an injective homomorphism fij : Γi ↪→ Γj and Ũi can be embedded in Ũj such that the following diagram commutes. -