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We use poset stratifications to construct a Grothendieck ring for stratifiable spaces. We then use it to compute the topological Euler characteristic of a number of constructions in topology and geometry. The computations are remarkably streamlined. Main applications pertain to orbit spaces and to spaces stratified by configuration spaces, like graph configuration spaces, orbit configuration spaces, or finite subset spaces. Our K0 can be viewed as an analog of the ring of constructible functions in the theory of Euler calculus, valid for a larger class of spaces. |