  It is proved that the isometry classes of pointed connected complete
Riemannian $n$-manifolds form a Polish space, $\mathcal{M}_*^\infty(n)$, with
the topology described by the $C^\infty$ convergence of manifolds. This space
has a canonical partition into sets defined by varying the distinguished point
into each manifold. The locally non-periodic manifolds define an open dense
subspace $\mathcal{M}_{*,\text{lnp}}^\infty(n)\subset\mathcal{M}_*^\infty(n)$,
which becomes a $C^\infty$ foliated space with the restriction of the canonical
partition. Its leaves without holonomy form the subspace
$\mathcal{M}_{*,\text{np}}^\infty(n)\subset\mathcal{M}_{*,\text{lnp}}^\infty(n)$
defined by the non-periodic manifolds. Moreover the leaves have a natural
Riemannian structure so that $\mathcal{M}_{*,\text{lnp}}^\infty(n)$ becomes a
Riemannian foliated space, which is universal among all sequential Riemannian
foliated spaces satisfying certain property called covering-continuity.
$\mathcal{M}_{*,\text{lnp}}^\infty(n)$ is used to characterize the realization
of complete connected Riemannian manifolds as dense leaves of
covering-continuous compact sequential Riemannian foliated spaces.
