Interval¶
An order-convex subset containing every ambient element between any two of its members.
Core Idea¶
An interval is a subset of a linearly ordered set that contains every ambient element between any two of its members. Real \([0,1]\) and integer \(\{2,3,4,5\}\) both pass this no-gap test under their respective orders. The rule, not bracket notation, defines the object.[ref-683fbdc409e6][ref-df31046d02bb]
Scope of Application¶
Real intervals include open, closed and half-open bounded sets, rays and singletons; the empty set is included by some conventions. Nonempty real intervals are exactly the connected subsets of \(\mathbb R\) with its usual topology, supporting intermediate-value reasoning. Integer intervals remain order-convex but need not be topologically connected.[^ref-683fbdc409e6]
Clarity¶
The ambient order matters. \(\{2,4\}\) is not an interval in \(\mathbb Z\) because 3 lies between its members but is absent. The symbol \(\infty\) in a ray's notation indicates an unbounded direction, not an included real endpoint.[^ref-df31046d02bb]
Manages Complexity¶
One closure condition replaces a list of bracket cases and makes operations easier to analyze. Intersections preserve interval status; a union can leave a gap. Interval Arithmetic is a separate method for computing with interval enclosures, not part of the interval's identity.[^ref-df31046d02bb]
Abstract Reasoning¶
Specify the ordered universe and a proposed subset. For every included pair, test whether every ambient point between them is included. One missing middle point refutes interval status. Apply connectedness theorems only after verifying that the setting is the real line with its usual topology.[ref-683fbdc409e6][ref-df31046d02bb]
Knowledge Transfer¶
The no-gap test transfers from real numbers to integers and other linear orders. An interval presupposes an Order relation but is a subset, not the ordering operation. Familiar endpoint forms and the real connectedness equivalence do not automatically transfer; Interval Arithmetic is a different operation on these objects.[ref-683fbdc409e6][ref-df31046d02bb]
[^ref-683fbdc409e6]: Leinster, University of Edinburgh Topology notes, §C2. [^ref-df31046d02bb]: University of Toronto order/topology notes, exercise 10.2.
Relationships to Other Abstractions¶
Current abstraction Interval Domain-specific
Parents (1) — more general patterns this builds on
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Interval presupposes Order Prime
Order intervals presuppose an order relation.
Hierarchy paths (3) — routes to 3 parentless roots
- Interval → Order → Comparison → Self Checking
- Interval → Order → Relation
- Interval → Order → Set and Membership
Neighborhood in Abstraction Space¶
Interval sits in a sparse region of the domain-specific corpus (84th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Formal Models & Logical Foundations (33 abstractions)
Nearest neighbors
- Filtration (algebra) — 0.82
- Ordered geometry — 0.82
- Ring Ideal — 0.82
- Linear order — 0.81
- Subspace Topology — 0.81
Computed from structural-signature embeddings · 2026-10-08