I am studying foliations of manifolds and at the moment I am stuck with trying to prove that the relation of coherence of foliated atlases is an equivalence relation.
I shall follow Candel and Conlon's Foliations I, but I want a different proof than theirs, as I shall explain in a moment. The definitions we need are the following. Let $X$ be a $d$-dimensional manifold. A foliated chart is a chart $(U,\varphi)$ such that $\varphi(U)=B_\tau \times B_\pitchfork$, for some $k$-dimensional open ball $B_\tau$ and $d-k$-dimensional open ball $B_\tau$.
A plaque of $(U,\varphi)$ is a set $P=\varphi^{-1}(B_\tau \times \{y\})$ for some $y \in B_\pitchfork$.
Two foliated charts $(U,\varphi),(V,\phi)$ are coherently foliated whenever, for each plaque $P$ of $U$ and $Q$ of $V$, the intersection $P \cap Q$ is open in both $P$ and $Q$.
A foliated atlas is an atlas of foliated charts of $X$ which are pairwise coherently foliated and two foliated atlases are coherently foliated whenever their union is coherently foliated, or equivalently, each chart in the first atlas is coherently foliated with each chart in the second.
Now, proving reflexivity and symmetry is straightforward since it is true for pair of charts, but proving transitivity is a bit trickier. The way they do it is to show that the definition of coherence of charts has an alternative characterization: the independence of the $y$ coordinates in $B_\pitchfork$ of the transition of coordinates $\phi \circ \varphi^{-1}$ from the $x$ coordinates in $B_\tau$.
Using this alternative characterization, it becomes quite easy, but I would like to prove transitivity using the fact that $P \cap Q$ is open in both $P$ and $Q$ for every plaques directly.
To prove transitivity, I need only prove the following statement.
Let $\mathcal A$ be a foliated atlas and let $(U,\varphi)$ and $(V,\phi)$ be foliated charts that are coherently foliated with every chart of $\mathcal A$. In this case, $(U,\varphi)$ and $(V,\phi)$ are coherently foliated.