# Definition:Subdivision (Real Analysis)/Normal Subdivision

## Definition

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

Let $P = \left\{{x_0, x_1, x_2, \ldots, x_{n - 1}, x_n}\right\}$ form a (finite) subdivision of $\left[{a \,.\,.\, b}\right]$.

$P$ 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:

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

## Historical Note

The name normal subdivision has been specifically coined for $\mathsf{Pr} \infty \mathsf{fWiki}$, as there appears to be no standard name for this concept in the literature.