Chapter 2 is an introduction to stratified spaces. We begin with filtered spaces and move progressively through more and more constrained classes, including manifold stratified spaces, locally cone-like spaces, the CS sets of Siebenmann, recursive CS sets, and topological and piecewise linear (PL) pseudomanifolds. To facilitate this last definition, we provide some background on PL topology. In the later sections of the chapter, we turn to some more specialized topics, including normalization of pseudomanifolds, pseudomanifolds with boundary, and other more specialized types of spaces, such as Whitney stratified spaces, Thom–Mather stratified spaces, and homotopically stratified spaces. We observe that the class of pseudomanifolds includes many spaces that arise naturally in other mathematical areas, such as singular varieties and orbit spaces of group actions. We also discuss stratified maps between stratified spaces and close with two specialized topics: intrinsic filtrations and products and joins of stratified spaces.

We motivate intersection homology theory by discussing how Poincaré duality fails on spaces with singularities. We see that one difficulty is the failure of general position, explaining why generalizations of general position will play an important role in the definition of intersection homology, which is a variant of simplicial or singular homology that recovers a version of Poincaré duality for singular spaces. We also discuss some conventions that will hold throughout the book and provide a quick overview of the difference between GM and non-GM intersection homology. We also provide a chapter-by-chapter outline of the rest of the book.