The theory of topological spaces decomposed into a locally finite partition of smooth manifold pieces (strata) arranged by incidence relations, together with control data (tubular or conical neighborhood structures, regularity conditions) that govern how strata meet and how invariants extend across singularities.

Principle

Organize a singular space by a stratification satisfying frontier and local-conicality/regularity conditions so that local models reduce to product/cone types and global invariants (intersection homology, constructible sheaves) can be defined and computed respecting the stratified structure.

Demonstration

An algebraic variety over R or C can be stratified by smooth manifolds according to multiplicity or singular locus; for example, a nodal plane curve decomposes into its smooth arcs and crossing points, and intersection homology computed relative to that stratification records how cycles meet the singularities.

Misapplication

Treating an arbitrary decomposition as a stratification without verifying frontier and local regularity (e.g., allowing wild, nonlocally finite pieces) invalidates many theorems and invariants; assuming a global product structure of neighborhoods when only a local cone exists leads to incorrect deductions.

Consequence

When stratified correctly, one obtains tools to extend Poincaré duality (intersection forms), define signature and L-classes, compute constructible sheaf cohomology, and control categories of stratified maps and cobordisms useful for singularity classification and topological invariants.

Reversal

The reversal is the manifold viewpoint that ignores stratification and treats the space only on its smooth strata, losing information about incidence and how singularities constrain global topology; alternatively, studying the singular locus alone focuses on the defects rather than the stratified whole.

Boundary

Applies to spaces admitting a locally finite, well-behaved stratification (Whitney, Thom‑Mather conditions or similar) and to contexts where conical/tubular control is given; it excludes arbitrary wild decompositions, measure-theoretic singularities without local manifold structure, and many fractal-like sets without manifold strata.

Semantic Tension

Tension exists between stratified-space theory and related notions such as orbifolds, which have mild quotient singularities, or CW stratifications and filtrations: nearby meanings differ in regularity, allowed isotropy and the algebraic/geometric tools applicable.

Synthesis

Stratified space theory decomposes singular spaces into manifold strata with controlled incidence and local conical models so that intersection-type invariants and sheaf-theoretic tools can be defined and used to classify singularities and extend manifold invariants across singular loci.