Definition:Subdivision (Real Analysis)/Finite

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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 $\closedint a b$.


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:

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


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


Some sources do not define the concept of infinite subdivision, and so simply refer to a finite subdivision as just a subdivision.


Sources