Intuitive reason why the Euler characteristic is an alternating sum? The Euler characteristic of a topological space is the alternating sum of the ranks of the space's homology groups.  Since homeomorphic spaces have isomorphic homology groups, however, even the non-alternating sum of the ranks of the homology groups is an invariant.  So, is there an intuitive reason why the Euler characteristic should be defined as an alternating sum instead of a non-alternating sum -- aside from the fact that other theorems, such as Gauss-Bonnet, would break (or at least need to be re-worked)?  Is this related to rank being additive?  If so, then a historical question: what would have been the original motivation of Euler to consider an alternating sum?

I wrote a blog post addressing this question here. In that post I show that this alternating behavior (a simpler but less general version of it coming from counting "cells") is forced by a few natural axioms, the most important of which is an inclusion-exclusion axiom
$$\chi(A \cup B) = \chi(A) + \chi(B) - \chi(A \cap B)$$
where some restrictions on $A$ and $B$ are necessary. I also ask for homotopy invariance in that post, but now I think this is actually unnecessary. 
