  We study the completeness and ultracompleteness numbers of a convergence
space. In the case of a completely regular topological space, the completeness
number is countable if and only if the space is $\v{C}$ech-complete, and the
ultracompleteness number is countable if and only if the space is
ultracomplete. We show that the completeness number of a space is equal to the
pseudopaving number of the upper Kuratowski convergence on the space of its
closed subsets, at $\emptyset$. Similarly, the ultracocompleteness number of a
space is equal to the paving number of the upper Kuratowski convergence on the
space of its closed subsets, at $\emptyset$.
