Brouwer homeomorphism theorem

Brouwer’s homeomorphism theorem states that in a two-dimensional, flat, infinite plane, if a continuous one-to-one transformation (called a homeomorphism) maps the plane onto itself, then that transformation must be equivalent to rotating, flipping, or shifting the plane—nothing more complicated. In other words, the plane’s shape and structure aren’t fundamentally changed by such transformations. It highlights that the plane's topology (its inherent connectedness and continuity) restricts the kinds of transformations possible, ensuring they’re all "simple" in a topological sense.