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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 ?
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