Many definitions of a singularity of a manifold $X^n$ are concentrating on the defining equations of it and the vanishing of the (partial) derivatives. My questions:
- What if $X^n$ isn't algebraic (so there are no defining equations) ?
- Is there a (useful) definition of a singulaity or a singular subset when $X^n$ is merly a topological manifold, but not differential ?