is locally detectable, i.e. for all continuous functions , is in iff for every there exist an open neighborhood and such that ,
let be elements of and a smooth function. Then is in .
Local detectability is equivalent to requiring that is an algebra of global sections of a given subsheaf of the sheaf of all continuous functions on ; in particular germs at every point can be defined.
For a manifold . For a differentiable space a tangent space can be defined at each . Define . By construction, decomposes into a disjoint union . There is an induced stratifold structure on the topological subspace , which we denote by .
A -dimensional stratifold is a differential space such that