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Finite unions of intervals are simple sets that are used quite often, e.g. in measure theory. (The construction of the Cantor set is a noble example). I realised that I do not have a name for them. Is there one? (Pluri-intervsals, multi-intervals, broken intervals,..?)

edit. I am pleased to see that this question is of some interest, thank you all for your comments. Why don't we give answers, each with a single proposal of name, so to see the most voted choices? (Shall I make this community wiki?)

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    $\begingroup$ The analog of "finite intervals" is common in some contexts, despite the illogic. For example, a "finite gap potential" is a certain potential whose spectrum has finitely many gaps (= intervals not in the spectrum). $\endgroup$
    – Christian Remling
    Commented May 27 at 19:36
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    $\begingroup$ There is no canonical name, but you can always call anything like that "a simple set" for the purposes of one particular exposition (at least, that is what I do in my measure theory courses). You may want to be a bit careful about whether you allow open, closed, or half-open intervals, etc, not to run into a conflict with your own definitions but otherwise, if used frequently and consistently, such terminology is easy to absorb and to follow. $\endgroup$
    – fedja
    Commented May 27 at 20:42
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    $\begingroup$ Pluri-interval is used in the (old, 1950 circa) Italian school of mathematics (Picone, Miranda, Caccioppoli, Fichera etc.) to denote cartesian product of intervals. By using analogy to simple functions in Lebesgue integration theory, why do not call unions of intervals "simple sets"? $\endgroup$
    – Daniele Tampieri
    Commented May 28 at 8:59
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    $\begingroup$ Such sets are key to o-minimal structures, but are never given a name when studied. Perhaps one could be very cheeky and refer to such sets as o-minimal. $\endgroup$
    – Calliope Ryan-Smith
    Commented May 28 at 9:30
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    $\begingroup$ @Lspice not exactly: a set is a finite union of intervals iff its characteristic functionis a step function . With simple functions we would get all measurable sets.... $\endgroup$
    – Pietro Majer
    Commented May 29 at 2:19

4 Answers 4

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The discussion in the comments has triggered my curiosity: I've done a little bit of research and I almost immediately stumbled upon the Wikipedia entry on the Peano-Jordan measure. There I found exactly the terms "simple sets" and "polyrectangles" used as synonyms for the unions of finite families of $n$-rectangles, i.e. for the set $$ S=\bigcup_{i=1}^q C_i\quad q\in\Bbb N_{>0} $$ where $n$ is the dimension of the Euclidean $\Bbb R^n$ considered and $C_i$ are $n$-rectangles defined as $C_i=[a_{1i}, b_{1i})\times\cdots\times[a_{ni}, b_{ni})$. However, I was not able to find where these sets are called in this way within the references stated in the relevant section of the entry: thus I started searching in my personal library. I found the locution "plurirectangles" (plurirettangoli in Italian) in the classical monograph [1], chapter II, §2.7, p. 119, as shown in the picture below:

enter image description here

This makes me sufficiently confident in proposing the adoption of this, somewhat traditional, terminology (i.e. plurirectangle, see the discussion in the comments by Pietro Majer and Daniele Tampieri).

Addendum. As I customarily do, when I try to be faster sacrificing precision I end up being at best badly inaccurate. The standard locution for finite unions of $k$-intervals is pluriintervals (or plurintervals if you prefer to translate so the Italian word plurintervalli): both Gaetano Fichera in his only book ([1a], chapter I, §I.12, p. 29 footnote (9)) and Carlo Miranda in his last (and recent) treatise ([2a], chapter II, §36, p. 160) use exactly this word, as shown in the following pictures: enter image description here

enter image description here

Possibly, this settles the problem.

Reference

[1] Federico Cafiero (1959), Misura e integrazione [Measure and integration] (Italian), Monografie matematiche del Consiglio Nazionale delle Ricerche 5, Roma: Edizioni Cremonese, pp. VII+451, MR0215954, Zbl 0171.01503.

Addendum references

[1a] Gaetano Fichera, Lezioni sulle trasformazioni lineari. I.: Introduzione all’analisi lineare (Lectures on linear transformations, I.: Introduction to linear analysis), (Italian) (third reprint, 1962) Trieste: Istituto Matematico dell'Università, pp. XIX+502 (1954), MR67346, Zbl 0057.33601.

[2a] Carlo Miranda, Istituzioni di analisi funzionale lineare. Volume I (Foundations of linear functional analysis. Volume I), (Italian) Unione Matematica Italiana. Bologna: Pitagora Editrice, pp. iii+596 (1978), Zbl 0697.46001.

[2b] Carlo Miranda, Istituzioni di analisi funzionale lineare. Volume II. (Foundations of linear functional analysis. Volume II), (Italian) Unione Matematica Italiana. Bologna: Pitagora Editrice. pp. 597-748 (1979), Zbl 0697.46002.

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    $\begingroup$ @M.G. thank you! $\endgroup$
    – Daniele Tampieri
    Commented May 29 at 6:22
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    $\begingroup$ Then what is your proposal? Plurirectangle is fine to me (I use it too in fact) but do you mean to use it in dim=1 too? (I woud be hesitant). Or did you mean to corroborate the choice of plurinterval by analogy with plurirectangle? At least in Italian, it seems well attested indeed. $\endgroup$
    – Pietro Majer
    Commented May 31 at 7:05
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    $\begingroup$ @PietroMajer, Cafiero uses plurirectangle for a general $k\in\Bbb N_{>0}$ (he uses the concept in his treatment of the Peano-Jordan measure). Plurinterval is more a synonym for $k$-rectangle: nevertheless Picone and Viola define also multidimensional intervals as rectangles, thus I think that pluriinterval wouldn't be a bad choice. But again I am in favour of plurirectangle for any general $k\in\Bbb N_{>0}$, just to keep on with the tradition. Well. my two cents. $\endgroup$
    – Daniele Tampieri
    Commented May 31 at 8:00
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    $\begingroup$ Sure. Using “pluri-interval” or “plurinterval” for $n=1$, but also for every dimension, is fine to me and indeed it is customary. On the contrary, “pluri-rectangle“ was expressly thought in dim n, so although in particular it should be ok for the case n=1, I think here thinking “x” as “a particular case of a generalisation of x “would be unnecessary, and a bit weird $\endgroup$
    – Pietro Majer
    Commented May 31 at 13:27
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    $\begingroup$ @PietroMajer, after analyzing your comments, I did a little more research and added what I found to my answer. $\endgroup$
    – Daniele Tampieri
    Commented Jun 3 at 21:59
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People working with conformal nets use the term
multi-interval to denote finite disjoint unions of intervals.

https://www.ms.u-tokyo.ac.jp/~yasuyuki/klm3.pdf

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As I see it, it would be preferable a simple term not referring to more advanced characterisations of more general objects (“Finite homology subset”? “Semialgebraic subset”? “Submanifold with boundary”? Also “pluri-rectangle” seems a particular case of an unnecessary generalisation). “Simple set” is indeed simple enough and has some analogy with “simple functions”, but somehow too generic (like other widely used terms as “normal”, “regular”, etc). Yet it seems to fit very well as “local term”, within a particular exposition (see fedja’s comment above). Among the given suggestions I like the most the self-explaining “multi-interval” and “pluri-interval”; “poly-interval”, is a neologism of mixed Greek-Latin formation, which I like less). The prefixes “multi-” and “pluri-“are almost synonimous. But here I like more “plures”, the comparative degree of “multi”, because it vaguely suggests the idea of adding more separated terms “in series”, starting from the case of one. On the contrary, “multi” seems often associated with the idea of co-presence of many entities “in parallel” (multitask, multilingual, multiple personality), even in mathematics (multigraph, multiplicity, multiset, multifunction).

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    $\begingroup$ My mind was set on 'multi-' up until you made that quite convincing argument that 'pluri-' is actually better! $\endgroup$
    – M.G.
    Commented May 31 at 13:41
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For a non-standard terminology:

In my own analysis lecture notes (sorry, not in a good enough stage to share too widely on the internet at large), I defined a tile in $\mathbb{R}^n$ to be a set of the form $[a_1,b_1)\times [a_2,b_2) \times \cdots \times [a_n,b_n)$, and I called a subset of $\mathbb{R}^n$ tileable if it can be written as a finite union of tiles.

(I only used half-open intervals since the point is to motivate and define Riemann integrals; this has the side benefit that the set of tileable sets are closed under union, intersection, set difference, and symmetric set difference. I suppose you can get the same if you include all four forms of intervals, but the notation gets a bit messy.)

I originally wanted to use the word "brick", taking my cue from Mikusinski in the book The Bochner Integral. However, "brickable" sounds a bit strange, at least compared with "tileable"; and the use of the word "tile" as a verb seems a bit more common compared to the use of the word "brick" as a verb.

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    $\begingroup$ Or, at least, of the word "brick" as this verb (sense 1); I think senses 4 and 5 are reasonably common. (My brain immediately leaps to puns, like, could I call the closure of a brick a "brick-a-bracket", because it is written as a product of $[a, b]$s?) $\endgroup$
    – LSpice
    Commented May 29 at 20:10

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