Refinement of a partition

Refinement of a partition
Tag	`0EEKF7YFBW`
URI	`m2://0EEKF7YFBW`
Kind	definition
Writing	W0 stub
MSC	26A42
Defines	Refinement of a partition
Uses	Partition of an interval

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Definition

Let $P$ and $Q$ be partitions of $[a, b]$ . The partition $Q$ is a refinement of $P$ if each point of $P$ is also a point of $Q$ .

Remarks

Any two partitions $P$ and $Q$ have a common refinement, made of the points of both. Refining a partition lowers the upper Darboux sum and raises the lower Darboux sum, or leaves them unchanged.

Explanations

L1 human

A refinement cuts the interval at more points. Each old subinterval is cut into one or more new subintervals.

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Partition of an interval definition
Refinement of a partition definition

Sources

Web https://en.wikipedia.org/wiki/Partition_of_an_interval, Refinement of a partition

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