# Homology on Pointed Sets 2

We wish to address four of the Eilenberg-Steenrod axioms for this category.  First note that for dimension we have $H(x_0)=x_0$ where $x_0$ denotes the trivial pointed set.

If $(X,x_0)$ and $(Y,y_0)$ are pointed sets, then we define the intuitive product pointed set as the pointed set $(X\times Y,(x_0,y_0)).$  If $X$ and $Y$ are also differential sets with differentials $d_1$ and $d_2,$ then we define the product differential component-wise: $d_1\times d_2 (x,y)=(d_1(x),d_2(y)).$  This is clearly a differential on $X\times Y,$ which we in turn call the product differential set.

If we then define the relation $\sim$ on $X\times Y$ where $(x,y)\sim (x',y')$ iff $(d_1(x),d_2(y))=(d_1(x'),d_2(y')),$ then we have $[(x,y)]_{d_1\times d_2}=[x]_{d_1}\times [y]_{d_2}.$  Hence we have

$\begin{array}{lcl}H(X\times Y)&=&[(x,y)]-(d(X\times Y)-(x_0,y_0))\\&=&[x]\times[y]-\left((d_1(X),d_2(Y))-(x_0,y_0)\right)\\&=&\left([x]-(d_1(X)-\{x_0\}),[y]-(d_2(Y)-\{y_0\})\right)\\&=&H(X)\times H(Y).\end{array}$

We can similarly define a formal sum of pointed spaces $\sqcup_i(X_i,x_i)$ whose corresponding homology is just defined as

$H\left(\bigsqcup_i(X_i,x_i)\right)=\bigsqcup_iH(X_i).$

So we have two forms of additivity.  We also have excision, for if $(X,A)$ is a pair of sets with $A$ a sub-differential set of $X$ and $U\subseteq A,$ then

$\begin{array}{lcl}H(X-U,A-U)&=&[x_0]\cap\left(X-U-\left((A-U)-\{x_0\}\right)\right)-d\left(X-U-(A-U)\right)\\&=&[x_0]\cap\left(X-(A-\{x_0\})\right)-d(X-A)\\&=&H(X,A).\end{array}$

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 $A$).  Let $f:(X,A)\to (Y,B)$ be a map between paired sets.  We say this is a paired set morphism if $f(A)\subseteq B.$  If we furthermore have that these are paired pointed sets $(X,A,x_0)$ and $(Y,B,y_0)$ with $A$ and $B$ sub-pointed sets, a paired set morphism $f$ between them is a paired pointed morphism if we also have that $f(x_0)=y_0.$  If both are differential sets as well, we can define a paired differential morphism if we further have $f\circ d_1=d_2\circ f.$  Let $f$ be a paired differential morphism between such sets.  Then we would like for its restriction $f:H(X,A)\to H(Y,B)$ 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 $f,g:(X,A)\to (Y,B)$ to be homotopic mod $A$ provided they agreed on $H(X,A).$