# Can you name this systems of subsets?

Recently I was working on systems of subsets satisfying the following properties

Obviously, it is just useful partitioning for further magic but I wonder if there is some theoretical treatment of such systems. Any suggestion for reading is welcome

• $4$ is not really a property, rather a definition of more variables... Jun 21, 2021 at 13:15
• It would improve your Question to highlight early on that $S$ is a topological space. Jun 21, 2021 at 17:59
• What is $\Sigma$ supposed to be? . Jun 21, 2021 at 22:36
• There is a relation between this and a thing called a Partition Of Unity. Jun 21, 2021 at 22:49
• $\Sigma$ should be switchning manifold. In fact, this partition comes from piece-wise smooth ODEs Jun 22, 2021 at 8:36