Euler calculus on spaces homeomorphic to definable sets and some applications

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.