  We show that integration with respect to the Euler-Poincar\'e characteristic
can be extended from the setting of definable sets to the setting of
topological spaces homeomorphic to definable sets. We use that extension to
generalize a result regarding sensor networks due to Ghrist and Baryshnikov, in
order to make it more flexible in applications. Finally, we obtain both an
extension and a combinatorial proof of a classical result about the
Euler-Poincar\'e characteristic of fiber bundles.
