This chapter establishes some notational conventions and fundamental constructions. It is not comprehensive, nor is all of it even necessary (the unnecessary bits are meant to supply context), and we assume the reader is familiar with most of it already. Some of the material presented in this chapter is redundant in the sense that it will be revisited later. For instance, the cone, wedge, and suspension of a space will be discussed later in terms of colimits, a perspective more in line with the philosophy of this book. The essential topics presented here which are utilized elsewhere are topologies on spaces and spaces of maps, homotopy equivalences, weak equivalences, and a few properties of the class of CW complexes whose extra structure we will need from time to time. Some familiarity with homotopy groups (mostly their definition) will also be useful, and to a much lesser extent some exposure to homology. Many proofs are omitted, and references are given instead. We will clarify which is which along the way.
Most references given in this chapter are from Hatcher's Algebraic topology [Hat02]. There are a few other modern references which the authors have found useful, and which contain most, if not all, of these preliminary results as well, such as [AGP02, Gra75, May99, tD08] (we especially like [AGP02] since it seems to be the most elementary text which follows this book's philosophy; [Gra75] is neither modern nor in print, but still a unique and valuable resource). We owe all of these sources a debt, in this chapter and elsewhere.
Spaces and maps
A topological space is a pair (X, τ), where τ is a collection of subsets of X, the members of which are called open sets, which contains both the empty set and X, and which is closed under finite intersections and arbitrary unions. However, it is customary to suppress the topology from the notation, so we simply write X in place of (X, τ), and typically denote generic topological spaces using capital Roman letters. A subbase for a topology τ on X is a subset of τ for which every element of τ is a union of finite intersections of elements in the subset; it is a sort of generating set for τ.