Skyrmions are important topologically non-trivial fields characteristic of models spanning scales from the microscopic to the cosmological. However, the skyrmion number can only be defined for fields with specific boundary conditions, limiting its use in broader contexts. Here, we address this issue through a generalized notion of the skyrmion derived from the de Rham cohomology of compactly supported forms. This allows for the definition of an entirely new \({\coprod }_{i=1}^{\infty }{{\mathbb{Z}}}^{i}\)-valued topological number that assigns a tuple of integers \(({a}_{1},\ldots,{a}_{k})\in {{\mathbb{Z}}}^{k}\) to a field instead of a single number, with no restrictions to its boundary. To demonstrate the power of our new formalism, we focus on the propagation of optical polarization fields and show, both theoretically and experimentally, that our newly defined generalized skyrmion number significantly increases the dimension of data that can be stored within the field while also demonstrating strong robustness. This novel topological number supports the idea that optical skyrmions are topologically protected states when interpreted in this more general context, and addresses some of the core issues in the application of optical skyrmions to communications and computing.
