Subintervals

Subintervals are the building blocks inside an interval. In real analysis, if I is an interval on the real line, a subinterval J is any interval contained in I. If I = [a,b], then every subinterval J is determined by two endpoints c and d with a ≤ c ≤ d ≤ b, and J can be one of [c,d], (c,d), [c,d), or (c,d], depending on whether its endpoints are included.

End points of a subinterval must lie within the end points of the original interval, and a subinterval may be equal to the original interval. When J ≠ I, it is called a proper subinterval. Degenerate cases are possible: [c,c] is a valid subinterval (a single point), while (c,c) is empty in the usual real-number convention.

Subintervals can be open, closed, or half-open, reflecting which endpoints are included. They preserve the property of being an interval and are connected subsets of the real line. The concept generalizes to any interval in the real numbers and to other contexts where one considers subsets that are themselves intervals.

In applications, subintervals are used to partition an interval, such as in defining Riemann sums or integrals, where an interval [a,b] is divided into a finite collection of subintervals. Subintervals also arise in analysis and topology when discussing continuity, limits, and subspace structures.

Examples include subintervals of [0,5] such as [1,3], (0,5], [0,0], and [2,2]. Each is contained within the original interval and satisfies the defining endpoint conditions.