Interval partition is a crucial concept in the field of integrals in real analysis. A partition of a closed and bounded interval is, by definition, a strictly increasing sequence of numbers starting from the initial point of the interval and reaching to the final point of the interval.

#### Definition of interval partition

An interval partition of a closed and bounded interval \(I = [a,b]\) is a strictly increasing sequence of real numbers having the following form: $$a = x_0 < x_1 < \cdots < x_n = b.$$ Every interval of the form \([x_{i-1} , x_{i}]\) is a subinterval of the interval partition \(\{x_i\}_{i=0}^{n}\) of the interval \(I = [a,b]\).

**Example**. Set \(x_i = \displaystyle \frac{i-1}{n}\) where \(1 \leq i \leq n+1\). Then, $$0 = x_0 < x_1 < \cdots < x_n = 1$$ is an interval partition of the interval \([0,1]\).

#### An alternative definition of interval partition

Two intervals are, by definition, almost disjoint if their intersection has at most one element.

**Examples**. In the following, we give some examples for almost disjointness:

- The intervals \([0,1]\) and \([2,3]\) are almost disjoint (in fact, they are disjoint).
- The intervals \([0,1]\) and \([1,2]\) are almost disjoint though not disjoint because their intersection is the singleton \(\{1\}\).
- The intervals \([0,2]\) and \([1,3]\) are not almost disjoint.

Let \(I\) be a nonempty, closed, and bounded interval. An interval partition of \(I\) is a finite collection $$\{I_1, I_2, \dots, I_n\}$$ of almost disjoint, nonempty, closed, and bounded subintervals of \(I\) whose union is \(I\).

#### Norm of an interval partition

The norm of an interval partition is the length of the longest subinterval of the given interval partition. In the language of mathematics, if $$a = x_0 < x_1 < \cdots < x_n = b$$ is an interval partition of \(I = [a,b]\), then its norm, denoted by \(\Vert \{x_i\}_{i=0}^{n} \Vert\), is $$\Vert \{x_i\}_{i=0}^{n} \Vert = \max\{\vert x_i – x_{i-1} \vert\}_{i=1}^{n}.$$

For example, the norm of the interval partition \(\left\{x_i = \displaystyle \frac{i-1}{n} \right\}\) of the interval \([0,1]\) is \(\displaystyle \frac{1}{n}\).

#### Refinement of an interval partition

Let $$a = x_0 < x_1 < \cdots < x_m = b$$ and $$a = y_0 < y_1 < \cdots < y_n = b$$ be interval partitions of the interval \(I = [a,b]\). Then, the interval partition \(\{y_i\}_{i=0}^{n}\) is a refinement of the interval partition \(\{x_i\}_{i=0}^{m}\) if \(m \leq n\) and \(\{x_i\}_{i=0}^{m} \subseteq \{y_i\}_{i=0}^{n}\).

#### Tagged partition

A tagged partition of a closed and bounded interval \(I = [a,b]\) is an interval partition $$a = x_0 < x_1 < \cdots < x_n = b$$ together with a finite sequence of real numbers $$t_0 < t_1 < \cdots < t_{n-1}$$ such that $$x_i \leq t_i \leq x_{i+1},$$ for each \(0 \leq i \leq n-1\).

#### The difference between interval partition and set partition

A set partition of a set \(S\) is a collection of some subsets \(\{S_i\}_{i \in I}\) of the set \(S\) with the following conditions:

- Each element of the set partition is nonempty, i.e., \(S_i\) is nonempty for each \(i \in I\).
- The union of all elements of the collection of the subsets in the set partition is the set \(S\), i.e. \(\bigcup_{i \in I} S_i = S\).
- Each two distinct elements of the collection of the subsets in the set partition are disjoint, i.e. if \(i\) and \(j\) are in \(I\) with \(i \neq j\), then \(S_i \cap S_j = \emptyset\).

Now, it is clear that the concept of interval partition is slightly different from the concept of set partition. However, they have also some similarities in this sense that if $$a = x_0 < x_1 < \cdots < x_n = b$$ is an interval partition of the interval \(I = [a,b]\) and we set \(S_i = [x_i , x_{i+1}]\), then we have the following:

- Each \(S_i\) is nonempty.
- The union of the \(S_i\)s is equal to the whole interval \(I\).

Though the third condition in the definition of set partition is not satisfied since $$S_i \cap S_{i+1} = \{x_{i+1}\}$$ while in the concept of set partition distinct subsets are disjoint.

The concept of interval partition is strongly dependent of the concept of order in the field of real numbers. For more, see the post on the properties of real numbers.