Semi-partition or pre-partition

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

1. $X = \overline{\bigcup\limits_i F_i}$ if $X$ is a topological space, or
2. $\mu\left(X\setminus\bigcup\limits_i F_i\right) = 0$ 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.

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Are these defintions useful in your work in some way so that you wanna know if there are any results about it or are you just wondering? Although I think it is a quite cool way of partitioning a set. =) +1 – Patrick Da Silva Aug 11 '11 at 12:37
@Patrick: I just use them to describe one example - and it is crucial for me that it is not a classical partition. – Ilya Aug 11 '11 at 12:45
As for the second case: in probability theory the term would be "partition almost everywhere". – Hagen Knaf Aug 11 '11 at 13:50
In the topological case I’d simply call it a (pairwise) disjoint family whose union is dense in $X$; I’ve not seen any special term for it. In fact, I can remember seeing it only once: such a family figures in the proof that almost countable paracompactness, a property once studied at some length by M.K. Singal, is a property of every space. – Brian M. Scott Aug 12 '11 at 20:06
@BrianM.Scott Would you be interested in writing some version of your comment as an answer so that this is no longer on the unanswered list? – Mark S. Jan 1 '14 at 16:24

In the topological case I'd simply call it a (pairwise) disjoint family whose union is dense in $X$; I've not seen any special term for it.