Let X be a topological space, $\mathcal{U} = \{U_\alpha\}_\alpha$ an open cover of $X$ and $\mathcal{F}$ a presheaf of abelian groups on $X$. Then one can define the Čech cohomology groups of $\mathcal{U}$ with values in $\mathcal{F}$: \begin{equation} \check{H}^k(\mathcal{U}, \mathcal{F}) \end{equation} The Čech cohomology groups of $X$ with values in $\mathcal{F}$ are usually defined by the inductive limit over the set of open covers ordered by refinement: \begin{equation} \check{H}^k(X, \mathcal{F}) = \varinjlim_{\mathcal{U}} \check{H}^k(\mathcal{U}, \mathcal{F}) \end{equation}
But I am confused with the notion of open cover. Indeed one can consider the open cover as a set $\{U_\alpha\}_\alpha\subset \mathcal{P}(X)$ but also as a map $A\to \mathcal{P}(X)$ for some index set $A$. The open cover as a set is the range of the open cover as a map. But the distinction is real, since in the second case you can consider repeated open sets in the cover.
It seems to me that it is the second notion that is used to define Čech cohomology but then one cannot talk about "the set of open covers ordered by refinement" and then take the inductive limit since in the second case you don't even have a set of open overs, but rather a class.
So which alternative is used for open covers?
I found my answer in A Gentle Introduction to Homology, Cohomology, and. Sheaf Cohomology by Jean Gallier and Jocelyn Quaintance, on page 238. As they said: "Most textboook presentations of Čech cohomology ignore this subtle point". They propose two answers, one by J.P Serre and one by Godement.
The idea is just to consider the second notion but to restric the index set so you get the first notion when you take the indutive limit.