Riemann's mapping theorem

Riemann's mapping theorem states that any simply connected, open, and proper subset of the complex plane (like a patch of the plane without holes) can be smoothly transformed into a standard shape, specifically a unit disk, through a conformal map. This transformation preserves angles and local shapes, allowing complex regions to be studied and understood by comparing them to the well-understood unit disk. Essentially, it guarantees that complex shapes without holes can be "flattened" or "mapped" onto a circle, simplifying analysis without distorting their fundamental geometric properties.