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I am reading Serre's Faisceaux Algébriques Cohérents (Henceforth FAC) and he uses some terminology I have not seen. I have searched around a bit but can't get a clean and clear definition.

Question: For $mathfrak{U}$ an open cover of a topological space, what is the definition of $mathrm{dim}(mathfrak{U})$?

The proof in which the terminology appears does seem to provide some clues:

Clue / Context: Here is the proposition and proof where the notion is used.

Corollary. $H^{q}(mathfrak{U},mathscr{F}) = 0$ for $q > mathrm{dim}(mathfrak{U})$.

By the definition of $mathrm{dim}(mathfrak{U})$ we have $U_{i_0}
cap cdots cap U_{i_q} = emptyset$ for $q >
mathrm{dim}(mathfrak{U})$, if the indices $i_0,dots,i_q$ are
distinct; hence $C^{'q}(mathfrak{U},mathscr{F})=0$, which shows that
$$H^{q}(mathfrak{U},mathscr{F}) = H^{'q}(mathfrak{U},mathscr{F})
= 0.$$

It appears to be the largest number of overlapping sets?

The above selection is from this English translation, and can be found on page 25.

Many of my tags are related to the context of the document my question comes from, not the question itself. Feel free to edit this if you find it inappropriate.

You are right, Serre uses the phrase "By the definition of $mathrm{dim}(mathfrak{U})$" but actually nowhere defines $mathrm{dim}(mathfrak{U})$ in FAC.

So we should take the explanation after the above phrase as the definition. That is, $mathrm{dim}(mathfrak{U})$ is the unique number in $mathbb{N} cup { infty }$ with the property that for all $q in mathbb{N}$ we have $q > mathrm{dim}(mathfrak{U})$ if and only if $U_{i_0} cap cdots cap U_{i_q} = emptyset$ for any choice of distinct indices $i_0,dots,i_q$. Equivalently, we have $q le mathrm{dim}(mathfrak{U})$ if and only if there exist distinct indices $i_0,dots,i_q$ such that $U_{i_0} cap cdots cap U_{i_q} ne emptyset$.

This shows that
$$mathrm{dim}(mathfrak{U}) = \ sup {q in mathbb{N} mid text{There exist distinct indices } i_0,dots,i_q text{ suich that } U_{i_0} cap cdots cap U_{i_q} ne emptyset} .$$

You have to be cautious. Serre takes the limit of a direct system of abelian groups indexed by open covers $mathfrak{U}$. There is no corresponding concept of a direct system of open covers.

Serre introduces a pre-ordering on the set $mathcal{C}$ of open covers by defining $mathfrak{U} le mathfrak{V}$ if $mathfrak{U}$ is finer than $mathfrak{V}$, i.e. if each $U in mathfrak{U}$ is contained in some $V in mathfrak{V}$. For example, if $mathfrak{U} subset mathfrak{V}$, then $mathfrak{U} le mathfrak{V}$. But there are no reasonable "maps" between open covers, so we cannot form something like a limit of the system of open covers.

Intituively, a cover $mathfrak{V}$ is refined by adding to $mathfrak{V}$ "small" open sets and by throwing away "big" open sets from $mathfrak{V}$.

Moreover, the topology $mathfrak{T}$ of $X$ belongs to $mathcal{C}$, however it is not the finest open cover, but the coarsest. In fact, each $mathfrak{U} subset mathfrak{T}$, hence $mathfrak{U}$ is finer than $mathfrak{T}$, and the covers $mathfrak{U}$ such that $mathfrak{T}$ is finer than $mathfrak{U}$ are precisely those with $X in mathfrak{U}$. So we do not approach $mathfrak{T}$ by taking finer and finer covers.

As Lord Shark the Unknown explained, one can associate to each open cover $mathfrak{U}$ a simplicial complex $K(mathfrak{U})$. The collection of all these complexes forms a system whose "maps" are contiguity classes of certain ordinary simplicial maps $K(mathfrak{U}) to K(mathfrak{V})$ which arise for $mathfrak{U} le mathfrak{V}$. This is used to define the Cech homology groups and Cech cohomology groups of the space $X$.