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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 $$a\leq\lambda\leq b\quad\forall a\in A, \forall b\in B.$$


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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I can't think of having seen specific words for these things in treatments of Riemann integration-the separation element isn't hard to talk about as bounding $A$ above and $B$ below, and the explicit concept of separation isn't necessary to say the superior sums bound the inferior sums above. Is there anything specific you're having trouble understanding? – Kevin Carlson Aug 14 '12 at 18:02
@KevinCarlson: I don't have trouble understanding, but have difficulty talking on this site about concepts of which I don't know the names (see this answer). If they have no definite name, I could stay with this. – enzotib Aug 14 '12 at 18:10
Well, in the link you're just giving a single name to the $\sup$ of a set and the $\inf$ of its upper bounds. It may be I'm undereducated, but it would be easy for me to believe the English-language community hasn't seen a need for a specific word when it has essentially no more content than $\sup$. – Kevin Carlson Aug 14 '12 at 18:28
To elaborate on Kevin's comments, when discussing with other people, it's probably easiest to just say $\sup A \leq \inf B$ or $\sup A = \lambda =\inf B$. Even if there are standard terms for these ideas (which I am not aware of), it's clear that they are not universally known, and so the best way to make yourself understood is to just be explicit about what you mean, ideally in the most compact way possible. – Aaron Aug 14 '12 at 18:58
Separated sets have a different definition in English, but Dedekind cuts are very similar to your "unique separation elements". – Per Manne Aug 14 '12 at 20:49
up vote 3 down vote accepted

This post intends to remove this question from the Unanswered list.

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. – egreg Oct 8 '13 at 22:41

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