For the context, I have seen various definitions, like this:
Or this:
I would like to concentrate to the first one (Definition 19.) and understand this definition. I think mesh there is supremum of some metric stuff. However, I don´t know, what is "diam $U$". Could you answer this, please?
Also, have you seen this definition of mesh, or different definitions?
$\endgroup$ 52 Answers
$\begingroup$$$\color{magenta}{\text{Mesh is simply the length of the largest sub-interval.}}$$
$\color{red}{\text{Example}}$: If we divide the interval $[1,2]$ into sub-intervals $[1,1.5]$, $[1.5,2]$, $[2,3]$, then the mesh is equal to $1$ being the length of the longest (last in this case) sub-interval.
Note that, by length of $[x,y]$, we mean, $|y-x|$.
$\endgroup$ $\begingroup$$\mathrm{diam} U$ is short for diameter. So the mesh is the smallest number where all the diameters of things in $\mathcal{U}$ are less than it.
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