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.

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

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 $\set {x_0, x_1, x_2, \ldots}$ forms an infinite subdivision of $\closedint a b$.

Normal Subdivision

$P$ 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:

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

Higher Dimensions

Rectangle

Let $R = \closedint {a_1} {b_1} \times \dotso \times \closedint {a_n} {b_n}$ be a closed rectangle in $\R^n$.

Let:

$P = \tuple {P_1, \dotsc, P_n}$

where every $P_i$ is a finite subdivision of $\closedint {a_i} {b_i}$.

Then $P$ is a finite subdivision of the closed rectangle $R$.

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

Sources

• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): partition (of an interval)
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): subdivision (of an interval)