Cellular structures play an essential role in topology, analysis and geometry; they appear in the form of CW-complexes, simplicial sets and so on. The idea of this book is to give a unified treatment of their fundamental geometric and topological (in the sense of general topology) properties. As a common basis for their representation we have chosen the CW-complexes.

CW-complexes were formally introduced in the literature in 1949 by the great English mathematician John H.C. Whitehead. To appreciate better the depth and perception of Whitehead's ideas, it is worth looking back into the development of algebraic topology; on this trip through history we take Solomon Lefschetz as our Virgil. In his beautiful history of the early development of algebraic topology (see Lefschetz, 1970), Lefschetz shows us how homology was defined by Henri Poincaré – whom he calls the ‘Founder’ of algebraic topology – using spaces with a combinatorial structure; Lefschetz then points out the next stage in the development of the subject, namely the definition of homology for topological spaces and the introduction of the homotopy groups of spaces. What Whitehead did was to impose again a combinatorial structure on the spaces and to show how this leads to a much deeper insight into their homotopy groups. This and other particularly interesting properties of CW-complexes explain why their presence is felt throughout many branches of mathematics.