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A X I abelian group Ap+1 Ap+l axiom Let barycentric subdivision based on infinite Cech cohomology Cech groups Cech theory choose closed star closed subcomplex coboundary operator cocycle coefficient group cohomology theory defined cohomology theory satisfying compact Hausdorff spaces compact pairs compact spaces completing the proof construction contained continuity axiom continuous map Eilenberg and Steenrod element of 35 exactness axiom excision axiom factor group Hp(X function not necessarily groups and homomorphisms Hence Hm Xa homology homotopy axiom identity map Induced homomorphisms induces isomorphisms infinite cochains inverse limit inverse system isomorphism of Hp(X iVp+i(U kernel Lemma locally finite complexes locally zero natural homomorphism necessarily continuous neighborhood of AXp+1 op(X open covering p-cocycle p-functions right hand side set of points showing simplex singular theories star refinement subset system of compact Theorem topological space Ua,A xp-i Xp+l Zp(X