Before revision:  As an application of this isomorphism  we are able to give succint proofs of the characterization of the persistent homology of products and joins of metric spaces.
After revision:  As an application of this isomorphism we are able to precisely characterize the type of intervals that appear in the persistence barcodes of the Vietoris-Rips filtration of any compact metric space and also to give succinct proofs of the characterization of the persistent homology of products and joins of metric spaces.