In the first question, Spaces of (complete) separable metric spaces, I asked about the Gromov–Hausdorff metric. Here, I am asking about the weaker notion of Gromov–Hausdorff convergence for pointed metric spaces. This is defined e.g. in section 8.1 of A Course in Metric Geometry by Burago, Burago, and Ivanov.
I am again interested in those metric spaces which are separable; since these spaces are not distinguishable from their completions even with respect to the Gromov–Hausdorff metric, I will also assume that their metrics are complete.
FTP's counterexample to separability of the Gromov–Hausdorff metric for part (1c) of my previous question should also work here. That leaves me with the following questions about the coarser topology on the space of pointed separable metric spaces:
(2a) Is this space of spaces (locally) metrizable? If so, is it completely metrizable? If not, what separation properties does it have?
(2b) Is the space of spaces connected, or even path-connected? If so, what is known about its homotopy type? If not, what is known about the (path) component containing the compact spaces?