reduced singular homology and wedge product

Let be based spaces, which admit a pointed contractible neighbourhood of , and the reduced singular homology, then it holds for the wedge product

By definition it holds, that

Then for each , let be the wedge product. It is open by definition of the quotient topology and furthermore by wedge product of pointed contractible spaces as contractible space the chain map is a quasi-isomorphism (cf. singular homology as homotopy invariant functor)

Then applying the homology functor results in a commutative diagram

(trust me ?, as simplices in are sent to 0) Here is an isomorphism, as the constant map is part of a homotopy equivalence.

Hence it follows, that their cokernels are naturally isomorphic (cf. colim functor, isomorphisms under functors)

Therefore we get

using excision theorem for gives

and the homeomorphism

(cf. subspace of wedge product without point as coproduct) gives

applying singular homology preserves coproducts gives

again by excision, we get

and again by similar reasoning for the cokernel

TODO