  We define a new topology, weaker than the gap topology, on the space of
selfadjoint unbounded operators on a separable Hilbert space. We show that the
subspace of selfadjoint Fredholm operators represents the functor $K^1$ from
the category of compact spaces to the category of abelian groups and prove a
similar result for $K^0$. We define the spectral flow of a continuous path of
selfadjoint Fredholm operators generalizing the approach of
Booss-Bavnek--Lesch--Phillips.
