Reformulating homology theory from pairs to single spaces I want to convert chapter IX of Eilenberg-Steenrod book to an exposition in single spaces, reminiscent to usual introductory expositions of simplicial or singular homology, but for Cech homology. This is done in Hocking-Young but not in an inverse system and only done fully for finite polytopes.
I was wondering if, given a homology theory, there is any general way to go back from pairs to single spaces, such that at least induced homomorphisms make sense? Or perhaps, where could it fail? I apologize for such an open ended question, I would greatly appreciate any pointers to references. I am not sure how to proceed in doing this myself, and I worry there is a good reason why Hocking-Young did it on such elementary level. Thanks!
Added after the accepted answer: the question was not phrased well, of course the point a comparison of constructions of homology rather than functors.
 A: Your question is misleading, it seems to ask the following:

If we are given homology functors for pairs $H_*(X,A)$, how do we get homology functors for single spaces $H_*(X)$?

The answer is obvious: Take $H_*(X) = H_*(X,\emptyset)$.
But your comment makes clear that this is not what you want to know. You are not interested in the homology functors as the final outcome of some construction, but in the construction itself.
There is a plethora of homology theories satisfying the Eilenberg-Steenrod axioms, and some of them are based on the following type of construction:

*

*Define two functors associating to pairs and single spaces chain complexes $C_*(X,A)$ and $C_*(X)$.


*Take $H_n(X,A) = H_n(C_*(X,A))$ and $H_n(X) = H_n(C_*(X))$.
This works for singular homology and we moreover only need the "absolute" chain complex $C_*(X)$ and define the "relative chain" complex as $C_*(X,A) = C_*(X)/C_*(A)$.
For Cech homology such an approach does not work. Eilenberg and Steenrod start with coverings of pairs, then form the nerves of these coverings and finally take the direct limit of the (simplicial) homology groups of these nerves. This is a more sophisticated construction than that of singular homology.
Here are some quotations:

p. 234: One should be careful not to confuse the sets $Cov(X)$ and $Cov(X,\emptyset)$.
However one may regard $Cov(X)$ as the subset of $Cov(X,\emptyset)$ consisting
of coverings $\alpha$ indexed by $(V_\alpha, \emptyset)$.


p. 236: Beginning with 2.4 the discussion of this chapter was limited to
coverings of pairs. However the definitions and results may be duplicated
for coverings of spaces (without a distinguished subset) and to
indexed families of sets which are not necessarily coverings.


p. 245: The Cech homology and cohomology groups of $(X,A)$, $A$, and $X$ are
defined as limits of suitable systems of groups defined over the directed
sets $Cov(X,A)$, $Cov(A,\emptyset)$, and $Cov(X,\emptyset)$ respectively.

This shows that Eilenberg and Steenrod define indeed $H^{Cech}_n(X) = H^{Cech}_n(X,\emptyset)$; they do not separately consider pairs and single spaces. But of course one could perform an alternative construction based on $Cov(X)$ to get functors $H^{Cech'}_n(X)$. Eilenberg and Steenrod don't do it, but you can repeat everything for $Cov(X)$ instead of $Cov(X,\emptyset)$. This is somewhat tedious and I doubt you learn much if you are performing it explicitly.
Anyway, if you have done this, you can use the fact that $Cov(X)$ is cofinal in $Cov(X,\emptyset)$ (recall Definitions VIII 2.1 and IX 2.8 plus Corollary VIII 3.16.) to show that $H^{Cech'}_n(X) \approx H^{Cech}_n(X,\emptyset) = H^{Cech}_n(X)$ canonically.
