# Definition:Subdivision (Real Analysis)

## Definition

Let $\left[{a \,.\,.\, b}\right]$ be a closed interval of the set $\R$ of real numbers.

### Finite

Let $x_0, x_1, x_2, \ldots, x_{n - 1}, x_n$ be points of $\R$ such that:

- $a = x_0 < x_1 < x_2 < \cdots < x_{n - 1} < x_n = b$

Then $\left\{{x_0, x_1, x_2, \ldots, x_{n - 1}, x_n}\right\}$ form a **finite subdivision of $\left[{a \,.\,.\, b}\right]$**.

### Infinite

Let $x_0, x_1, x_2, \ldots$ be an infinite number of points of $\R$ such that:

- $a = x_0 < x_1 < x_2 < \cdots < x_{n - 1} < \ldots \le b$

Then $\left\{{x_0, x_1, x_2, \ldots}\right\}$ forms an **infinite subdivision of $\left[{a \,.\,.\, b}\right]$**.

## Normal Subdivision

$P$ is a **normal subdivision of $\left[{a \,.\,.\, b}\right]$** if and only if:

- the length of every interval of the form $\left[{x_i \,.\,.\, x_{i + 1} }\right]$ is the same as every other.

That is, if and only if:

- $\exists c \in \R_{> 0}: \forall i \in \N_{< n}: x_{i + 1} - x_i = c$

## Also known as

Some sources use the term **partition** for this, but the latter term has an alternative and more general definition so it is probably better not to use it.