I've been reading about strong (and weak) $C^r$-topologies on the space of $C^r$-maps between $C^s$-manifolds $M$ and $N$ ($s \ge r$) from the textbooks of Hirsch and Wall (both called Differential Topology), and I noticed that they use different definitions for the strong and weak $C^r$-topology.
Hirsch's definition: take a $C^r$-function $f: M \to N$, locally finite set of charts $\Phi = \{(U_i, \varphi_i)\}_i$ on $M$, a set of charts $\Psi = \{(V_i,\psi_i)\}_i$ on $N$, a family of compact sets $K = \{K_i\}_i$ with $K_i \subseteq U_i$, and a family of positive real numbers $\varepsilon = \{\varepsilon_i\}_i.$ Then a basic neighborhood of Hirsch's strong topology is \begin{multline} \mathcal{N}^r(f; \Phi, \Psi, K, \varepsilon) = \{g \in C^r(M,N) \textrm{ }| \textrm{ } (\forall i) \textrm{ }g(K_i) \subset V_i, \textrm{ } (\forall i, x \in \varphi_i(K_i), k = 0,\dots,r) \textrm{ } \\ ||D^k(\psi_i f \varphi_i^{-1})(x) - D^k(\psi_i g \varphi_i^{-1}(x)|| < \varepsilon_i\}. \end{multline}
Wall's definition: Wall's definition is simpler - for an open set $W \subseteq M \times J^r(M,N)$, a basis set is given by $$B(W) := \{f \in C^r(M,N) \textrm{ } | \textrm{ } \Gamma(j^rf) \subseteq W\}.$$ $\Gamma$ denotes the graph of a function. It is the minimal topology such that the injective map $j^r: C^r(M,N) \to C^0_F(M, J^r(M,N))$ is a topological embedding, where $C^0_F(M,J^r(M,N))$ denotes the fine topology.
In general, given topological spaces $X$ and $Y$, the fine topology is the one generated by the basis $B(W) = \{f \in C(X,Y) \textrm{ } | \textrm{ } \Gamma(f) \subseteq W\}$, where $W \subset X \times Y$ is an open set.
QUESTION: How would one prove that these two definitions are equivalent, i.e. generate the same topology?
I was able to prove equivalence for their definitions of weak topology, as well as one direction for the strong one - namely, that Hirsch's topology is no finer than Wall's. However, I'm having a hard time with the other direction - it just seems too complicated to construct all of the necessary data - the $\Phi$, $\Psi$, $K$ and $\varepsilon$... I provide what I've done below.
Hirsch $\subseteq$ Wall: By Lemma A.4.1 of Wall's book, if $X,Y$ are metric spaces, then the sets $$I(\{K_\alpha, U_\alpha\}) = \{f \in C(X,Y) \textrm{ } | \textrm{ } (\forall \alpha) \textrm{ } f(K_\alpha) \subseteq U_\alpha\},$$where $K_\alpha$ are compact and $U_\alpha$ open, and $\{K_\alpha\}$ is a locally finite collection, form a subbase for the fine topology on $C^0(X,Y).$
Now if I'm given a set $\mathcal{N}^r(f; \Phi, \Psi, K, \varepsilon)$, then it's equal to $(j^r)^{-1}(I(\{K_i,W_i\})$ where $W_i \subseteq J^r(U_i,V_i)$ is the inverse image by the jet chart of the set \begin{multline} W_i' := \{(\varphi_i(x),\psi_i g (x), D(\psi_i g \varphi_i^{-1})(x),\dots,D^r(\psi_i g \varphi_i^{-1})(x)) | x \in U_i, \forall k = 0, \dots, r : ||D^k(\psi_i g \varphi_i^{-1})(x) - D^k(\psi_i f \varphi_i^{-1})(x)|| < \varepsilon_i\} \end{multline} and therefore it's open in "Wall's strong topology."
As for the other direction, I look at some $f \in (j^r)^{-1}(I(\{K_\alpha, W_\alpha\}_{\alpha \in A}))$ - for every individual pair $(K_\alpha, W_\alpha)$, I can find a set $\mathcal{N}^r(f; \Phi_\alpha, \Psi_\alpha, K'_\alpha, \varepsilon_\alpha)$ where the collections $\Phi_\alpha, \Psi_\alpha, K'_\alpha, \varepsilon_\alpha$ are finite such that $$f \in \mathcal{N}^r(f; \Phi_\alpha, \Psi_\alpha, K'_\alpha, \varepsilon_\alpha) \subseteq (j^r)^{-1}(I(\{K_\alpha,W_\alpha\})).$$Therefore, a possible solution would be to take the intersection for all $\alpha \in A$ to get $$\mathcal{N}^r(f; \cup_{\alpha \in A} \Phi_\alpha, \cup_{\alpha \in A} \Psi_\alpha, \cup_{\alpha \in A} K'_\alpha, \cup_{\alpha \in A} \varepsilon_\alpha).$$However, I don't know if $\cup_{\alpha \in A} U_\alpha$ will be locally finite. Of course, I can take a locally finite refinement, but I don't know what I'd do with the other data - $\Psi$, $K'$, $\epsilon$.
PS: I am willing to provide more details in an edit or in private - also, it's possible that most of what I wrote is useless, so feel free to write a solution that doesn't rely on my ramblings. However, I'd like it to be as self-contained as possible, because many general-topological facts in Hirsch and Wall go unproven.
