Let $A$ be a set. Then it seems common to name *cover* of $A$ any set $\mathcal C$ of sets with $\bigcup{\mathcal C}\supseteq A$.

Is there then a good name for the particular covers $\mathcal C$ with $\bigcup{\mathcal C}=A$ ?

Are there standard names for those covers $\mathcal C$ with $\emptyset\notin\mathcal C$ ?

Question 2 also arises for partitions. Well-regarded books allowing the empty set to be a member of a partition (which I then would only call a *quasipartition*) but using the term differently later on include Kelley's and Dugundji's topology texts.