singular homology

The generalization of simplicial homology, throwing away the triangulation (so does the notion of complex-approximation) and the gluing rules (规则相处). Relatively, cellular homology seems to just loosen the gluing rules (characteristic map).

similarities: formal sum-freely generating; when preimage and image have the same dimension, close~closed loop, exact~not around holes.

Proof of equivalence: relative homology and five lemma

singular cohomology~for Abelian group, defined as cohomology of Eilenberg-Maclane spectrum