Homeomorphisms and Topological Invariants

In this chapter, we define those maps between two spaces which will make two spaces equivalent, hence indistinguishable from the topological point of view. Intuitively, these maps reflect processes of stretching and contracting. As we also demand that there can be no ‘sticking’ transformations, the maps have to be bijective. These mappings are the so-called homeomorphisms and they are defined as transformations between topological spaces that are bijective and the map as well as its inverse are continuous. The main role of homeomorphisms is the classification of topological spaces.