There are two notions of convergence of a sequence of metric space. One is by the Gromov Hausdorff distance for compact metric spaces, another one is the pointed Gromov Hausdorff convergence for pointed (possibly non-compact) metric spaces. We here suggest a third one, for sequences of complete (possibly non-compact) metric space.
Let $Y$ be a metric space, and $CL(Y)$ be the set of closed subsets of $Y$. We can equip $CL(Y)$ with the Vietoris topology. Let $\mathcal O\subset CL(Y)$ be a basic open set. Denote $\mathcal N(Y,\mathcal O)$ to be the set of all metric space $Z$ (up to isometry) such that there exists an isometric embedding $f$ of $Z$ into the interior of $Y$ with $f(Z)\in\mathcal O$. If I did not make mistake, it seems that we can use the collection of all possible $\mathcal N(Y,\mathcal O)$ (we run over all possible O and Y) as a basis, to define a topology $\tau$ on the set of all isometry classes of complete metric space. As a result, given a sequence of complete metric spaces, we can define the notion of convergence using $\tau$.
Is this definition already know? Can someone please suggest a reference? Moreover, is there a theorem for this topology $\tau$ analog to the Gromov compactness theorem?
By Taras’s comment: Given a sequence of complete separable metric space such that when embedded in the Urysohn universal space, their images converge under the Vietoris topology. Then does this garuntee that the sequence converges under $\tau$, and this limit space is isometric to the one in the Urysohn universal space?
Edit: I modified the definition to require an embedding of Z into the interior of Y, to allow convergence of a sequence of metric from outward, eg an decreasing sequence of set under inclusion.