  K-theory for $ \sigma$-C*-algebras (countable inverse limits of C*-algebras)
has been investigated by N. C. Phillips [{\it K-Theory} {\bf 3} (1989),
441--478]. We use his representable K-theory to show that the space of Fredholm
modular operators with coefficients in an arbitrary unital $ \sigma$-C*-algebra
$A$, represents the functor $X \mapsto {\rm RK}_{0}(C(X, A))$ from the category
of countably compactly generated spaces to the category of abelian groups.
