= Topological invariance of Euler characteristic
A <homeomorphism> preserves every <homology group>, hence the <Euler characteristic> wherever the finite rational <Betti number> sum is defined. In fact, <homotopy invariance of homology> makes <Euler characteristic> invariant under <homotopy equivalence>. Thus different finite <CW complex> structures on the same space have equal alternating cell counts.<Algebraic Topology by Allen Hatcher>[Algebraic Topology, §2.2, p. 146, homotopy-type independence immediately before Theorem 2.44]{ref}
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