  We show that a large class of two-field models of single-bubble open
inflation do not lead to infinite open universes, as it was previously thought,
but to an ensemble of very large but finite inflating `islands'. The reason is
that the quantum tunneling responsible for the nucleation of the bubble does
not occur simultaneously along both field directions and equal-time
hypersurfaces in the open universe are not synchronized with equal-density or
fixed-field hypersurfaces. The most probable tunneling trajectory corresponds
to a zero value of the inflaton field; large values, necessary for the second
period of inflation inside the bubble, only arise as localized fluctuations.
The interior of each nucleated bubble will contain an infinite number of such
inflating regions of comoving size of order $\gamma^{-1}$, where $\gamma$
depends on the parameters of the model. Each one of these islands will be a
quasi-open universe. Since the volume of the hyperboloid is infinite, inflating
islands with all possible values of the field at their center will be realized
inside of a single bubble. We may happen to live in one of those patches of
comoving size $d\lesssim \gamma^{-1}$, where the universe appears to be open.
In particular, we consider the ``supernatural'' model proposed by Linde and
Mezhlumian. There, an approximate U(1) symmetry is broken by a tunneling field
in a first order phase transition, and slow-roll inflation inside the nucleated
bubble is driven by the pseudo-Goldstone field. We find that the excitations of
the pseudo-Goldstone produced by the nucleation and subsequent expansion of the
bubble place severe constraints on this model. We also discuss the coupled and
uncoupled two-field models.
