Equisingularity
In algebraic geometry, an equisingularity is, roughly, a family of singularities that are not non-equivalent and is an important notion in singularity theory. There is no universal definition of  equisingularity but Zariki's equisingularity is the most famous one. Zariski's equisingualrity, introduced in 1971 under the name " algebro-geometric equisingularity", gives a stratification that is different from the usual Whitney stratification on a real or complex algebraic variety. See also 
stratified space

References

Further reading 
https://mathoverflow.net/questions/299314/a-general-definition-of-an-equisingular-family-of-singular-varieties

algebraic geometry