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An italian real analysis book frequently uses the following definitions

Definition 1. Separated sets, separation elements.

The sets $A,B\subset\mathbb{R}$ are said to be separated if $a\leq b\quad\forall a\in A, \forall b\in B.$ It is said a separation element every $\lambda\in\mathbb{R}$ such that $a\leq\lambda\leq b\quad\forall a\in A, \forall b\in B.$

and

Definition 2. Adjacent sets.

The separated sets $A,B$ are said to be adjacent if they have a unique separation element.

These definitions are used, for example, to introduce the Riemann integral as the unique separation element between the sets of the inferior integral sums and the superior integral sums, when it exists.

My question is that I cannot be able to find such definitions on English language Wikipedia, so I would to know if they are used and under which names they go.

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    Separated sets have a different [definition](http://en.wikipedia.org/wiki/Separated_sets) in English, but [Dedekind cuts](http://en.wikipedia.org/wiki/Dedekind_cut) are very si$m$ilar to your "u$n$ique separation elements".2012-08-14

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Consensus in the comments suggests that there are no standard English terms for the definitions you describe.

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    Just to add some information: in Italian, which seems to be also the OP's mother tongue, it's common to use "separati" for the first definition and "contigui" for the second one. I don't know whether "contiguous" is used in English in this context, but it surely would render the idea.2013-10-08