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Comrnnxssz MULTIPLICATIONS AND INTERSECTIONS Frxm Euzumwrs Mam
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acyclic Betti numbers carrier chain-homotopic chain-mapping chains closed sets closed subcomplex cocycles coeﬂicients coﬁnal cohomology combinatorial compact compactum components consequence contains coordinates corresponding coset deﬁned deﬁnition denote diam dimensions discrete disjoint dual categories duality theorems dually paired elements equivalent Euclidean complex Euclidean space ﬁeld ﬁnd ﬁnite complexes ﬁnite number ﬁnite open coverings ﬁrst ﬁxed point follows group G Hausdorff space hence Hilbert parallelotope holds homology groups homology theory homomorphism homotopy I K I identiﬁed implies incidence numbers induces inﬁnite integral intersection isomorphism Kronecker index Let G likewise locally ﬁnite manifolds mapping merely mesh metric metric space multiplication n-sphere neighborhood notations open sets operator orientable p-cycle polyhedral complex projection proof properties prove reﬁnement relation replaced satisﬁes simple simplexes simplicial complex subgroup subset Suppose T1-space topological group topological space transformation vector spaces vertex vertices Vietoris X X Y zero zero-cycle