TITLE: For which classes of topological spaces Euler characteristics is defined?
QUESTION [10 upvotes]: 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, the Euler characteristics of an open interval according to this definition is $-1$.
This definition rases a second (maybe obvious) question. 
Question 2. Suppose $X$ is a locally compact space whose 1 point compactification is a $CW$ complex, 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 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.

REPLY [2 votes]: The Euler Characteristic can be defined for a larger class of spaces. For example, Euler Characteristic is defined for definable sets in an o-minimal system such as: semi-linear sets, semi-algebraic sets, algebraic varieties, compact smooth manifolds... and not all of them are locally compact. 
However, when they are not locally compact, then it is not necessarily true that the different definitions of the characteristic give the same result. If you are interested in this point of view you can read:

L. Van den Dries. Tame topology and o-minimal structures. Vol. 248. London Mathematical Society Lecture Notes Series. Cambridge University Press, 1998.
J. Curry, R. Ghrist y M. Robinson. “Euler calculus with applications to signals and sensing”. Proceedings of Symposia in Applied Mathematics. Vol. 70. 2012, pages: 75-146. (they have a pdf freely and legally available online here)