A universal Riemannian foliated space

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