"Interval" as a technical term in mathematics is a connected subset of a linearly (= totally) ordered set.

As such, the interval inherits the order. And, indeed, it does not have to be enumerable as the interval [0, 1] of the real numbers shows. Also, note that the empty set is not connected, so {} is not a proper interval. Another non-interval (of the natural numbers) is {0, 2, 3} because it is not connected (as 1 is missing).

Cartesian products of intervals are usually not intervals (for some canonical order of the product set), so one shouldn't call things like {1, 2, 3} x {0, 1, 2} x {0} intervals. The name cuboid would be much better.

-- Marc

Am Sa., 15. Dez. 2018 um 03:16 Uhr schrieb John Cowan <xxxxxx@ccil.org>:

On Fri, Dec 14, 2018 at 8:04 PM Per Bothner <xxxxxx@bothner.com> wrote:

A "range" is a sequence - it can be treated as a space-optimized immutable vector.
(In contrast an "interval" is a set - there is no inherent order.)

I would say that an interval has order; what it doesn't have is enumerable elements.

--
John Cowan http://vrici.lojban.org/~cowan xxxxxx@ccil.org
There is / One art / No more / No less
To do / All things / With art- / -Lessness --Piet Hein

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Prof. Dr. Marc Nieper-Wißkirchen

Universität Augsburg
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