# Euler characteristic for non-compact manifolds

How can one generalize the Euler characteristic to non-compact manifolds?
Furthermore, is there a way to generalize the notion of an intersection number to non-compact manifolds, so that one could use the definition of the Euler characteristic as the self-intersection number of the diagonal?

References for further reading would also be appreciated.

For example, the Euler characteristic of $\mathbb{R}^n$ is $1$, but the compactly supported Euler characteristic is $(-1)^n$. (In particular it is not a homotopy invariant.) Note that the Euler characteristic of the sphere, which is the one-point compactification of $\mathbb{R}^n$, is $1 + (-1)^n$; one reason to prefer the compactly supported Euler characteristic is that it has better additivity properties.