I'm trying to get familiar with Grothendieck topology. The following is taken from FGA Explained:
Definition 2.45. Let $\mathcal{C}$ be a category. $\{U_i\to U\}_{i\in I}$ is a set of arrows. A refinement $\{V_a\to U\}_{a\in A}$ is a set of arrows such that for each index $a\in A$ there is some index $i\in I$ such that $V_a\to U$ factors through $U_i\to U$.
Definition 2.47. Given two topologies $\mathcal{T}$ and $\mathcal{T'}$ on $\mathcal{C}$, we say that $\mathcal{T'}$ is subordinate to $\mathcal{T}$ and write $\mathcal{T'}\prec \mathcal{T}$, if every covering in $\mathcal{T'}$ has a refinement that is a covering in $\mathcal{T}$.
Definition 2.52. If $\mathcal{T}$ is a topology of $\mathcal{C}$, the saturation $\overline{\mathcal{T}}$ of $\mathcal{T}$ is the set of refinements of coverings in $\mathcal{T}$.
I want to prove this claim (Proposition 2.53(v)):
Claim: A topology $\mathcal{T'}$ is subordinate to $\mathcal{T}$ if and only if $\mathcal{T'}\subseteq \overline{\mathcal{T}}$.
What I have tried: I tried to prove that every covering in $\mathcal{T'}$ is a refinement of a covering in $\mathcal{T}$, but this 'naive' attack doesn't seem to work. I think I should use the axioms of Grothendieck topology in some clever way, but from this point I'm stuck.
Any hints or references will help me a lot. If the statement of the question is unclear or incorrect, please let me know. Thank in advance!
I don't think this notion of saturation is right as stated: say $\mathcal{T}$ is a topology of $\mathcal{C}$, then since $\{U\to U\}$ is a covering in $\mathcal{T}$, any collection of arrows $\{U_i\to U\}$ will be a covering in $\overline{\mathcal{T}}$, since every $U_i\to U$ factors through $U_i\to U \to U$.
The correct definition is: the saturation $\overline{\mathcal{T}}$ consists of collections of arrows $\{U_i\to U\}$ which admit a refinement that is in $\mathcal{T}$. With this definition proposition 2.53 is easy.