  This paper is devoted to the space of unbounded Fredholm operators equipped
with the graph topology, the subspace of operators with compact resolvent, and
their subspaces consisting of self-adjoint operators. Our main results are the
following: (1) Natural maps between these four spaces and classical spaces of
bounded operators representing K-theory are homotopy equivalences. This
provides an alternative proof of a particular case of results of Joachim. (2)
The subspace of unbounded essentially positive Fredholm operators represents
odd K-theory. (3) The subspace of invertible operators in each of these spaces
of unbounded operators is contractible.
