R.L. Moore's theorem

R.L. Moore's theorem states that if you have two shape spaces (called manifolds) that are locally similar to Euclidean space, and a continuous, one-to-one, and onto function (called a homeomorphism) between them, then these spaces are essentially the same in shape. This means you can smoothly deform one into the other without tearing or gluing, confirming they have identical topological properties. In simpler terms, the theorem assures that certain complex shapes can be considered the same if they can be continuously transformed into each other, emphasizing the importance of local properties in understanding global shape equivalence.