Consider some metric spaces, then
[Hausdorff distance][1] and [Gromov-Hausdorff convergence][2]

Also, introduce categorical notation, it may be very useful.


  [1]: https://en.wikipedia.org/wiki/Hausdorff_distance
  [2]: https://en.wikipedia.org/wiki/Gromov%E2%80%93Hausdorff_convergence