On deformations of isolated singularity functions

We study multi-parameters deformations of isolated singularity function-germs on either a subanalytic set or a complex analytic spaces. We prove that if such a deformation has no coalescing of singular points, then it has constant topological type. This extends some classical results due to Lê & Ramanujam (1976) and Parusiński (1999), as well as a recent result due to Jesus-Almeida and the first author. It also provides a sufficient condition for a one-parameter family of complex isolated singularity surfaces in ${\text{C}}^3$ to have constant topological type. On the other hand, for complex isolated singularity families defined on an isolated determinantal singularity, we prove that $\mu$-constancy implies constant topological type.