  We construct globally hyperbolic spacetimes such that each slice $\{t=t_0\}$
of the universal time $t$ is a model space of constant curvature $k(t_0)$ which
may not only vary with $t_0\in\mathbb{R}$ but also change its sign. The metric
is smooth and slightly different to FLRW spacetimes, namely, $g=-dt^2+dr^2+
S_{k(t)}^2(r) g_{\mathbb{S}^{n-1}}$, where $g_{\mathbb{S}^{n-1}}$ is the metric
of the standard sphere, $S_{k(t)}(r)=\sin(\sqrt{k(t)}\, r)/\sqrt{k(t)}$ when
$k(t)\geq 0$ and $S_{k(t)}(r)=\sinh(\sqrt{-k(t)}\, r)/\sqrt{-k(t)}$ when
$k(t)\leq 0$. In the open case, the $t$-slices are (non-compact) Cauchy
hypersurfaces of curvature $k(t)\leq 0$, thus homeomorphic to $\mathbb{R}^n$; a
typical example is $k(t)=-t^2$ (i.e., $S_{k(t)}(r)=\sinh(tr)/t$). In the closed
case, $k(t)>0$ somewhere, a slight extension of the class shows how the
topology of the $t$-slices changes. This makes at least one comoving observer
to disappear in finite time $t$ showing some similarities with an inflationary
expansion. Anyway, the spacetime is foliated by Cauchy hypersurfaces
homeomorphic to spheres, not all of them $t$-slices.
