Definition:Subdivision (Real Analysis)

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Definition

Let $\left[{a \,.\,.\, b}\right]$ 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 $\left\{{x_0, x_1, x_2, \ldots, x_{n - 1}, x_n}\right\}$ form a finite subdivision of $\left[{a \,.\,.\, b}\right]$.


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 $\left\{{x_0, x_1, x_2, \ldots}\right\}$ forms an infinite subdivision of $\left[{a \,.\,.\, b}\right]$.


Normal Subdivision

$P$ is a normal subdivision of $\left[{a \,.\,.\, b}\right]$ if and only if:

the length of every interval of the form $\left[{x_i \,.\,.\, x_{i + 1} }\right]$ 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 this, but the latter term has an alternative and more general definition so it is probably better not to use it.