  Topological invariance of the intersection homology of a pseudomanifold
without codimension one strata, proven by Goresky and MacPherson, is one of the
main features of this homology. This property is true for codimension-dependent
perversities with some growth conditions, verifying $\overline p(1)=\overline
p(2)=0$. King reproves this invariance by associating an intrinsic
pseudomanifold $X^*$ to any pseudomanifold $X$. His proof consists of an
isomorphism between the associated intersection homologies
$H^{\overline{p}}_{*}(X) \cong H^{\overline{p}}_{*}( X^*)$ for any perversity
$\overline{p}$ with the same growth conditions verifying $\overline p(1)\geq
0$.
  In this work, we prove a certain topological invariance within the framework
of strata-dependent perversities, $\overline{p}$, which corresponds to the
classical topological invariance if $\overline{p}$ is a GM-perversity. We also
extend it to the tame intersection homology, a variation of the intersection
homology, particularly suited for ``large'' perversities, if there is no
singular strata on $X$ becoming regular in $X^*$. In particular, under the
above conditions, the intersection homology and the tame intersection homology
are invariant under a refinement of the stratification.
