Definition:Subdivision (Real Analysis)

This page is about Subdivision in the context of Real Analysis. For other uses, see Subdivision.

Definition

Let $\closedint a b$ be a closed interval of the set $\R$ of real numbers.

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 $\set {x_0, x_1, x_2, \ldots, x_{n - 1}, x_n}$ form a finite subdivision of $\closedint a b$.

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 $\set {x_0, x_1, x_2, \ldots}$ forms an infinite 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.

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

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 deprecated on $\mathsf{Pr} \infty \mathsf{fWiki}$.