  We study the gap (= "projection norm" = "graph distance") topology of the
space of (not necessarily bounded) self--adjoint Fredholm operators in a
separable Hilbert space by the Cayley transform and direct methods. In
particular, we show that the space is connected contrary to the bounded case.
Moreover, we present a rigorous definition of spectral flow of a path of such
operators (actually alternative but mutually equivalent definitions) and prove
the homotopy invariance. As an example, we discuss operator curves on manifolds
with boundary.
