Intervallne

Intervallne is a term used in fictional or speculative mathematical writings to denote a generalized form of the classical interval. In these contexts, an intervallne is a construct designed to explore how interval-like objects behave under nonstandard assumptions about endpoints, uncertainty, or multiplicity. It is not an established term in mainstream mathematics.

Definition: In the standard fictional framework, an intervallne on a totally ordered set (S, ≤) is described by a pair (L, R) of endpoint specifications, where L and R are nonempty sets of elements of S that describe the left and right boundaries. An element x belongs to the intervallne I if and only if there exist l ∈ L and r ∈ R with l ≤ x ≤ r. When L and R are singletons {a} and {b}, I reduces to the ordinary interval [a, b].

Properties: Intervallnes are convex and connected in the order topology if L and R are intervals themselves; they preserve monotone transformations under a boundary-preserving map. The theory typically treats end-boundary variability (uncertainty or fuzziness) by allowing L and R to be intervals or fuzzy sets, yielding a family of nested intervallnes.

Applications and reception: The concept often appears in discussions about generalized interval arithmetic, uncertainty representation, or as a teaching device to contrast ordinary intervals with generalized boundary conditions. As a coined term in fiction, it serves to illustrate how mathematical objects can be generalized in ways that retain core interval intuition while inviting new questions.

See also: Interval (mathematics), Order topology, Interval arithmetic, Fuzzy interval.