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