We wish to address four of the Eilenberg-Steenrod axioms for this category. First note that for dimension we have where denotes the trivial pointed set.
If and are pointed sets, then we define the intuitive product pointed set as the pointed set If and are also differential sets with differentials and then we define the product differential component-wise: This is clearly a differential on which we in turn call the product differential set.
If we then define the relation on where iff then we have Hence we have
We can similarly define a formal sum of pointed spaces whose corresponding homology is just defined as
So we have two forms of additivity. We also have excision, for if is a pair of sets with a sub-differential set of and then
We lastly address homotopy and omit the long exact sequence as we have not developed a dimensional concept (although recall we constructed a short exact sequence under an assumption on ). Let be a map between paired sets. We say this is a paired set morphism if If we furthermore have that these are paired pointed sets and with and sub-pointed sets, a paired set morphism between them is a paired pointed morphism if we also have that If both are differential sets as well, we can define a paired differential morphism if we further have Let be a paired differential morphism between such sets. Then we would like for its restriction to be well-defined. Hopefully some combination of requirements like normality and being a paired differential morphism will work, but I have not yet convinced myself. If it worked, we would in turn just define two maps to be homotopic mod provided they agreed on