Definition:Subdivision (Real Analysis)/Normal Subdivision
Definition
Let $\closedint a b$ be a closed interval of the set $\R$ of real numbers.
Let $P = \set {x_0, x_1, x_2, \ldots, x_{n - 1}, x_n}$ form a (finite) subdivision of $\closedint a b$.
$P$ is a normal subdivision of $\closedint a b$ if and only if:
- the length of every interval of the form $\closedint {x_i} {x_{i + 1} }$ 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 the concept of a subdivision.
However, the latter term has a different and more general definition, so its use is discouraged on $\mathsf{Pr} \infty \mathsf{fWiki}$.
Some use the term dissection, but again this also has a different meaning, and is similarly discouraged on $\mathsf{Pr} \infty \mathsf{fWiki}$.
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.