TITLE: What is the purpose of the category of topological pairs?
QUESTION [5 upvotes]: Apologies for the vague question.
In the beginning of his book "Algebraic topology" (1.3) Spanier says that for "technical reasons" he considers not just the homotopy category but rather the larger homotopy category of pairs.
I see that some of the definitions and results may be formulated for pairs, but I fail to see its real purpose.
What is/are the fundamental advantage/s of the category of topological pairs over the usual category of topological spaces?

REPLY [5 votes]: Two basic concepts of algebraic topology are homotopy groups and homology groups. In both cases it does not suffice to consider "absolute spaces" $X$.
In the case of homotopy groups we need a basepoint $x_0 \in X$ to introduce the fundamental group $\pi_1(X,x_0)$ and the higher homotopy groups $\pi_n(X,x_0)$ for $n > 1$. But okay, here we only consider special pairs of the form $(X,\{x_0\})$.
The standard approach to homology theory (as you can find it in Spanier's book and most other textbooks) is to define not only the homology groups $H_n(X)$ of spaces, but also the relative homology groups $H_n(X,A)$ of pairs $(X,A)$.  Only this ingredient allows to formulate the exactness axiom and the excision axiom (see Spanier section 4.8). Without these homology axioms (or if you want, these properties of homology) you cannot calculate homology groups $H_n(X)$ of spaces, not even of simple spaces like the spheres $S^k$.
Singular homology satisfies these axioms. For excision see Spanier's Corollary 4.6.5, for exactness Theorem 4.5.4. Note that the latter is a theorem concerning chain complexes which implies the exactness axiom for singular homology. But surprisingly Spanier does not state this as an explicit theorem, he only mentions it in the text preceding Lemma 4.5.9 and once again after he axiomatically defined the concept of a homology theory.
I should not keep secret that there are alternative approaches to axiomatic homology theory which do not use relative homology groups. See for example Hatcher's "Algebraic Topology" section 2.3. But even in these approaches you need the category of pairs of spaces to formulate the axioms.
Conclusion: Algebraic topology without using pairs would be fairly unproductive.