This page is about Subdivision in the context of Real Analysis. For other uses, see Subdivision.
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:
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 numberofpoints of $\R$ such that:
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:
That is, if and only if:
Let $R = \closedint {a_1} {b_1} \times \dotso \times \closedint {a_n} {b_n}$ be a closed rectangle in $\R^n$.
Let:
where every $P_i$ is a finite subdivision of $\closedint {a_i} {b_i}$.
Then $P$ is a finite subdivision of the closed rectangle $R$.
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 $\mathsf{Pr} \infty \mathsf{fWiki}$.
