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 p(1) = 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 p * (X) ∼ = H p * (X * ) for any perversity p with the same growth conditions verifying p(1) ≥ 0.In this work, we prove a certain topological invariance within the framework of strata-dependent perversities, p, which corresponds to the classical topological invariance if 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. Definition 1.3. A stratified space is a filtered space such that any pair of strata, S andDefinition 1.4. Let X be a stratified space. The depth of X is the greater integer m for which it exists a chain of strata, S 0 ≺ S 1 ≺ · · · ≺ S m . We denote depth X = m.Example 1.5. If X is stratified, each construction of Example 1.2 is a stratified space.Definition 1.6. A stratified map, f : X → Y , is a continuous map between stratified spaces such that, for each stratum S ∈ S X , there exists a unique stratumObserve that a continuous map f : X → Y is stratified if, and only if, the pull-back of a stratum S ′ ∈ S Y is empty or a union f −1 (S ′ ) = ⊔ i∈I S i , with codim S ′ ≤ codim S i for each i ∈ I. Therefore, a stratified map sends a regular stratum in a regular one but the image of a singular stratum can be included in a regular one. Example 1.7. Let X be a stratified space. The canonical projection, pr : M × X → X, the maps ι t : X →cX with x → [x, t], ι m : X → M × X with x → (m, x) and the canonical injection of an open subset U ֒→ X, are stratified for the structures described in Example 1.2. Let us recall some properties of stratified maps from [4, Section A.2]. Proposition 1.8 ([4, Proposition A.23]). A stratified map, f : X → Y , induces the order preserving map (S X , ) → (S Y , ), defined by S → S f .Let us introduce the notion of homotopy between stratified maps. Here, the product X × [0, 1] is endowed with the product filtration. Definition 1.9. Two stratified maps f, g : X → Y are homotopic if there exists a stratified map, ϕ : X × [0, 1] → Y , such that ϕ(−, 0) = f and ϕ(−, 1) = g. Homotopy is an equivalence relation and produces the notion of homotopy equivalence between stratified spaces.The following notion of locally cone-like stratified space has been introduced by Siebenman, [20]. Definition 1.10. A CS set of dimension n is a filtered space, ∅ ⊂ X 0 ⊆ X 1 ⊆ · · · ⊆ X n−2 ...
