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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. =) +12011-08-11
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    @Patrick: I just use them to describe one example - and it is crucial for me that it is not a classical partition.2011-08-11
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    As for the second case: in probability theory the term would be "partition almost everywhere".2011-08-11
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    @Hagen: Thank you, I still wonder if there is a term concerning the first case since I use both conditions.2011-08-11
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    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.2011-08-12
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    If the sets $F_i$ were open, the system fulfilling condition 1 could perhaps be called [maximal](http://www.google.com/#q="maximal+cellular+family"+topological) [cellular family](http://www.google.com/#q="cellular+family"+topological).2011-08-14
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    @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?2014-01-01

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Brian M. Scott wrote:

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.