Question 1: In this setting, the Euler characteristic of a finite group or monoid G is 1/G, and the Euler characteristic of a finite ________ is the sum of 1/G_{i}, where we picked one representative group G_{i} for each connected component of the groupoid.  

Question 2: Multiple proofs, including their flaws and limitations, are used as examples in Proofs and Refutations by ________.  

Question 3: The Euler characteristic can be defined for connected ________ by the same V − E + F formula as for polyhedral surfaces, where F is the number of faces in the graph, including the exterior face.  

Question 4: A version used in ________ is as follows.  

Question 5: if X is a ________, and one uses Euler characteristics with compact supports, no assumptions on M or N are needed.  

Question 6: In mathematics, and more specifically in ________ and polyhedral combinatorics, the Euler characteristic (or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent.  

Question 7: The Euler class, in turn, relates to all other ________ of vector bundles.  

Question 8: For closed smooth manifolds, the Euler characteristic coincides with the Euler number, i.e., the ________ of its tangent bundle evaluated on the fundamental class of a manifold.  

Question 9: Another generalization of the concept of Euler characteristic on manifolds comes from ________.  

Question 10: The concept of Euler characteristic of a bounded finite poset is another generalization, important in ________.  

