On the topology of families of isolated singularities

We study the topology of analytic families of $n$-dimensional complex hypersurfaces having an isolated singularity at the origin. We prove that such a family is $\mu$-constant if and only if it admits an uniform Milnor radius, which happens if and only if it has constant embedded topological type. In particular, this solves the $\mu$-constant problem, formulated by D.T. L\^e and C.P. Ramanujam in 1976.