Let $a,b \in \mathbb{R}$ with $a<b$ A partition of an interval $[a,b]$ is a set of nonempty subintervals $\{ [a,x_1), [x_1,x_2), \dots , [x_{n-1}, b] \}$ for some positive integer $n$ That is, $a<x_1<x_2<\dots<x_{n-1}<b$ Note that $n$ is the number of subintervals in the partition.

Subinterval partitions are useful for defining Riemann integrals.

Note that subinterval partition is a specific case of a partition of a set since the subintervals are defined so that they are pairwise disjoint.