  We prove an extension to the simplicial Nerve Lemma which establishes
isomorphism of persistent homology groups, in the case where the covering
spaces are filtered. While persistent homology is now widely used in
topological data analysis, the usual Nerve Lemma does not provide isomorphism
of persistent homology groups. Our argument involves some homological algebra:
the key point being that although the maps produced in the standard proof of
the Nerve Lemma do not commute as maps of chain complexes, the maps they induce
on homology do.
