The following predicates are used in the definition of total boundedness of uniform spaces: a space is bounded if the predicate holds for every entourage, specifically for the first predicate (the second predicate is also related with total boundedness, the margins are too small to describe this relationship).
Let $(U;E)$ be a pair of a set $U$ and a binary relation $E$ on this set.
The first predicate: There exists a finite cover $S$ of $U$ such that $\bigcup_{A\in S}(A\times A)\subseteq E$.
The second predicate: There exist a finite set $B$ such that $\bigcup_{b\in B}E[b] = U$.
My question: Are there names (terms) for these predicates?