  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 L\^e \& Ramanujam
(1976) and Parusi\'nski (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 $\C^3$ 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.
