## Elementary Topology: A Combinatorial and Algebraic Approach |

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1-chain 1-cochain angle antipodal points appears ball boundary curve bounded called chords circle circuit closed curve coboundary colors complex condition Consider continuous coordinates corresponding covering cylinder defined definition derive describe determined direction disk distance distribution vector divided edges equal equation equivalent Euclidean Euclidean space Euler characteristic example faces feasible Figure finite fixed four function give hemisphere Hence identified infinity initial intersection Klein bottle letter lines locus manifold matrix Möbius band moves negative neighborhood occurrences opposite optimal orbits oriented origin pair particular plane pole polygons polyhedral solids positive potential problem projective plane prove replaced represented Riemann surface rotation satisfied shows simple single singularities solution sphere surface tangent terminal theorem three-dimensional three-space topological torus tree triangles values vector field vertex vertices zero