The cochain algebra C*(X;\(\Bbbk \))

As usual, we work over an arbitrary commutative ground ring, k. Recall from §3 that given two complexes (M, d) and (N, d) we can form the complex (Hom(M, N),d) with d(f) = df — (—1)deg fd. In particular, the normalized singular cochain complex of a topological space X is the cochain complex $$ C^* (X;\Bbbk ) = Hom\,(C_* (X;\Bbbk ),\Bbbk ). $$