In this note we investigate a
cohomology theory H#(X,G), defined by M. C. McCord, which is dual to a
homology theory based on hyperfinite chains of miscrosimplexes. We prove that if X
is a locally contractible, paracompact space then H#(X,G) ≃ Hč#(X,Hom(Z∗,G))
where Hč# is the Čech theory. Nonstandard analysis, particularly the Saturation
Principle, is used in this proof in essential way to construct a fine resolution of the
constant sheaf X ×Hom(Z∗,Z). This gives a partial answer to a question of
McCord. Subsequently, we prove a proposition from which it is deduced that
Hom(Z∗,Z) = {0} i.e. H#(X,Z) = {0} if X is paracompact and locally
contractible. At the end we briefly discuss a related cohomology theory which is
obtained by application of the internal (rather than external) Hom(⋅,G)
functor.
