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Another definition of a topological space
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algebraic arcwise connected barycentric coordinates barycentric subdivision base-point boundary called chain on F Chapter circular disc closed path condition connected space continuous mapping defined definition deformation denoted dimension element Euclidean simplex Euclidean space example Exercise finite number functions fundamental group geometrical given Hausdorff space Hint homologous modulo homologous to zero homology groups homomorphism homotopic with respect homotopy Hp(E Hr(E Hr(Kr Hv(E identity mapping inclusion mapping infinite cyclic integers interval isomorphism joining kernel lemma let F limit point linear combination linear mapping modulo F neighbourhood obtained open sets operation oriented simplex p-chain p-simplex pair paths based proof Prove quotient group r-dimensional real numbers relative cycle relative homology class represented satisfying sequence set of points shown simplicial complex simply singular simplex sphere subgroup subset subspace surface tetrahedron topological property topological space torus triangle tt(E union vertices xv x2