Eilenberg-Steenrod Axioms for covariant functors on $Top^2$ $Top^2$ is a category of all pairs of topological spaces $(X,A)$ where $A$ is a subspace of $X$. The morphism in this category is $(X,A)\to (X',A')$ with $Im(A)\subset A'$. Denote $\phi$ as the empty set. 
a) Why would one even consider pairs of topological spaces first place? Shouldn't $Top$ category is more natural to be considered instead?
There are 4 axioms required for covariant functors $G_n:Top^2\to Ab$.


*

*Requirement of homotopy axiom is clear as homology functor $H_n:Top\to Ab$ does not distinguish homotopies. 

*Exactness Axiom: Write $G_n(X,\phi)$. Any pair of $(X,A)$ has $i:(A,\phi)\to (X,\phi)$ and $j:(X,\phi)\to (X,A)$ with $i,j$ canonical injection maps. There is a long exact sequence $\cdots\to G_{n+1}(X,A)\to G_n(A,\phi)\to G_n(X,\phi)\to G_n(X,A)\to\cdots $
I think $j$ is also surjective map as $(X,\phi)\to (X,A)$ is basically $id_X:X\to X$ map. So that long exact sequence comes from $(A,\phi)\to (X,\phi)\to (X,A)$. 
b) Is $(X,\phi)$ treated as the projective object here?
c) Normally I would have long exact sequence induced from a short exact sequence in some $Ab$ category. $Top^2$ is not $Ab$ category. How should I understand the induced long exact sequence? Clearly I have initial object in $Top^2$ by $(\phi,\phi)$


*Excision Axiom. Given a pair $(X,A)$ and $U\subset X$ with $\bar{U}\subset interior(A)$, the inclusion $(X-U,A-U)\to (X,A)$ induces isomorphism $G_n(X-U,A-u)\to G_n(X,A)$ for all $n\geq 0$. 


d) Why would one want $\bar{U}\subset interior(A)$ rather than $U\subset interior(A)$? Where is isomorphism orginally coming from or what is the motivation that one wants isomorphism on $G_n(X-U,A-U)\to G_n(X,A)$? 


*Dimension Axiom. For every one-point space $P$, we have $G_n(P)=Z$ for $n=0$ $0$ other wise which is clear. 

 A: A) one motivation is the pair $(X,A)$ acts somewhat like the quotient $X/A$ and this quotient along with the subspace $A$ may be better understood then axiom 2 gives us a way to study $X$. For example when $A$ is such that $H_n(X,A)$ vanishes we have the isomorphism $H_n(A) \approx H_n(X)$.
B) $(X,\phi)$ is not necessarily projective here. If it were then for morphism $f:(X,\phi) \rightarrow (Y,\phi)$ and epimorphism $q:(Z,\phi) \twoheadrightarrow (Y,\phi)$ and $f$ factors through $q$. But any morphism in Top can be realized as a morphism of the form $(X,\phi ) \rightarrow (Y,\phi)$ in Top$^2$ so $X$ is projective in Top for all $X$ which is not necessarily true.
C) The long exact sequence is in Ab as we have the functor $G_n:\mathbf{Top}^2\rightarrow  \mathbf{Ab}$.
D) Excision can stated equivalently for subspaces $A,B \subset X$ whose interiors cover $X$ the inclusion $(B, A \cap B) \hookrightarrow (X,A)$ induces isomorphisms $G_n(B,A\cap B)\rightarrow G_n(X,A)$. The covering of $X$ by interiors seems more natural and is equivalent to the closure $\bar U \subset int \, A$.
