  For the minimally coupled scalar field in Einstein's theory of gravitation we
look for the space of solutions within the class of closed Friedmann universe
models. We prove that D = 1 or D > 1, where D is the (fractal) dimension of the
set of solutions which can be integrated up to t to infinity. (D > 0 was
conjectured by PAGE (1984)). We discuss concepts like ``the probability of the
appearance of a sufficiently long inflationary phase" and argue that it is
primarily a probability measure q in the space V of solutions (and not in the
space of initial conditions) which has to be applied. q is naturally defined
for Bianchi-type I cosmological models because V is a compact cube. The
problems with the closed Friedmann model (which led to controversial claims in
the literature) will be shown to originate from the fact that V has a
complicated non-compact non-Hausdorff Geroch topology: no natural definition of
q can be given. We conclude: the present state of our universe can be explained
by models of the type discussed, but thereby the anthropic principle cannot be
fully circumvented.
