For a given space $X$ the partition is usually defined as a collection of sets $E_i$ such that $E_i\cap E_j = \emptyset$ for $j\neq i$ and $X = \bigcup\limits_i E_i$.
Does anybody met the name for a collection of sets $F_i$ such that $F_i\cap F_j = \emptyset$ for $j\neq i$ but
- $X = \overline{\bigcup\limits_i F_i}$ if $X$ is a topological space, or
- $\mu\left(X\setminus\bigcup\limits_i F_i\right) = 0$ if $X$ is a measure space.
I guess that semipartition or a pre-partition should be the right term, but I've never met it in the literature.