Betti numbers

Betti numbers are topological invariants that quantify the shape and structure of a space. Specifically, the \(k\)-th Betti number measures the number of \(k\)-dimensional holes in a shape: - The 0th Betti number counts connected components, - The 1st counts loops or tunnels, - The 2nd counts voids or cavities, and so on. They help mathematicians understand the intrinsic features of complex shapes, regardless of how they are twisted or stretched, providing a systematic way to classify and analyze their fundamental connectedness and voids.