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. The answer of Algori indicates that a reasonably large class of spaces for which Euler characteristics can be defined are locally compact spaces $X$, whose one point compactification $\bar X$ is a CW complex. Then we can define $\chi(X)=\chi(\bar X)-1$. For example, should we conisder, that the Euler characteristics of an open interval according to this definition is $-1$? -1$. This definition rases a second (As far as I understand open interval maybe obvious) question.
Question 2. Suppose $X$ is not CW a locally compact space whose 1 point compactification is a $CW$ complex, at least according to wiki)and $Y$ is a subspace of $X$ such that both $Y$ and $X\setminus Y$ have this property. Is it ture that $\chi(X)=\chi(Y)+\chi(X\setminus Y)$?
Also, I was thinking, that Euler characteristics is more fundamental then homology. But homology.But can it be defined for spaces, where homology is not defined?
Finally, Quiaochu pointed out below that a very similar question was already discussed previously on mathoverflow.