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I was reading a book and it had the following sentence:

$A$ is a refinement of $B$

where $A$ and $B$ are sets.

What does this mean? Perhaps $A \subseteq B$ ?

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In the Context of partition it is clear what it means. – user17090 Dec 17 '11 at 10:51
"Refinement" usually has this meaning in topology. – Zev Chonoles Dec 17 '11 at 10:52
Perhaps adding a little more context (and maybe even the name of the book, chapter and perhaps also page number) could help. – Martin Sleziak Dec 17 '11 at 11:13
"Any partition α of a set X is a refinement of a partition ρ of X—and we say that α is finer than ρ and that ρ is coarser than α—if every element of α is a subset of some element of ρ. Informally, this means that α is a further fragmentation of ρ. In that case, it is written that α ≤ ρ." From: – gnometorule Dec 17 '11 at 12:05
@gnometorule: That is my tag. The original was both [set-theory] and [elementary-set-theory]. The two tags may coexist if there is a damn good reason. Until the OP returns with a reason (i.e. a reference) I would rather assume that it is an elementary problem. Moreover one can consider the possibility of refinement of topoligies as elementary just as well. – Asaf Karagila Dec 17 '11 at 13:28

Here are the two notions of refinement that come up most often in my work:

A topology $\tau$ on a set $X$ refines another topology $\sigma$ on $X$ if $\sigma\subseteq\tau$.

If $P$ and $Q$ are partitions (or covers) of a set $X$, then $P$ refines $Q$ if for all $U\in P$ there is $V\in Q$ such that $U\subseteq V$.

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