On (ultra-) completeness numbers and (pseudo-) paving numbers

Abstract

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 $\emptyset$.