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I don't know the term in english so I will only give its definition (the term in french is pavé).

It a subset P of $\mathbb{R^n}$ such that $P= I_1 \times I_2 \times ... \times I_n$. With $I_i$ an open interval or half-open interval.

My question is: is the empty-set a "pavé".

Or alternatively, is it an interval?

I know that in any topology, $\emptyset$ is open and closed, but is it an interval?

aribaldi
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    $\emptyset$ is, indeed, an interval. There are $11 $ types of intervals, you will find the list here. Go to the french page if you preferh. – Jean-Claude Arbaut Sep 29 '16 at 17:34
  • For the language thing see English word for a pavé in French – quid Sep 29 '16 at 17:36
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    This obviously depends on the definition of "interval," but it satisfies the "order" definition of an interval. Ultimately, whether you include it or not depends on whether you want to include it or not. Is it useful to treat the empty set as "pavé?" Does it make it more complicated to include it - would we find ourselves writing "If $S$ is an non-empty pavé set..." more often than "If $S$ is a pavé set..."? – Thomas Andrews Sep 29 '16 at 17:37
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    I'm more suggesting for the question of "pavé" than "interval." I believe the empty set should be called an interval. If "pavé" is used for the purposes similar to $n$-cell, then I don't think you want "pavé" to include empty sets. @Jean-ClaudeArbaut – Thomas Andrews Sep 29 '16 at 17:45

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