Definition:Subdivision (Real Analysis)/Infinite

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Definition

Let $\left[{a \,.\,.\, b}\right]$ be a closed interval of the set $\R$ of real numbers.


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 $\left[{a \,.\,.\, b}\right]$.


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.