By making use of a simple
connection with Banach algebras we introduce certain relations into singular
homology and cohomology at the chain level and show that we obtain homology and
cohomology theories. The deviation between singular and the new theory is measured
by what turns out to be another homology theory HM. One of the main results is
that HM is zero on simplicial complexes but not on metric spaces in general. This
shows that for any coefficient group there are an infinite number of different
homology theories agreeing with the associated homology theory on simplicial
complexes.
