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).
