I would like to know something more than what is written on wikipedia http://en.wikipedia.org/wiki/Euler_characteristic

What would be some large (largest?) class of topological spaces for which $\chi$ is defined, so that all standard properties hold, for example that $\chi(X)=\chi(Y)+\chi(Z)$ if $X=Y \cup Z$, ($Y\cap Z=0$).

ADDED. For example, should we conisder, that the Euler characteristics of an open interval is $-1$? (As far as I understand open interval is not CW complex, at least according to wiki). Also, I was thinking, that Euler characteristics is more fundamental then homology. But can it be defined for spaces, where homology is not defined?