Let $n \in \mathbb{N}^\ast$ and $$F = \{f \in \mathcal{C}([0,1]^n,\mathbb{R}) : f \text{ is convex and } f|_{\mathrm{int}(I)} \text{ is } C^1\}$$ where $\mathcal{C}([0,1]^n,\mathbb{R})$ is the set of continuous maps from $[0,1]^n$ to $\mathbb{R}$.
Is there a natural topology on this set ? I thought about the topology induced by the compact-open topology on $\mathcal{C}([0,1]^n,\mathbb{R})$ but I heard in some paper about the "$C^1$-compact-open topology" in the following framework: if we consider the set $$F' = \{f \in \mathcal{C}(\mathbb{R}^n,\mathbb{R}) : f \text{ is convex and } C^1\}$$ then the author describes it as the topology such that its subbase consists of all sets of the form $\{f \in F' : \forall x \in K, f(x) \in O \text{ and } \nabla_x f \in O'\}$, where $K \subset \mathbb{R}^n$ is compact, $O \subset \mathbb{R}$ is open and $O' \subset \mathbb{R}^n$ is open.
Is this a "classical" topology" ? Do you have some references ? What is its interest, compared to the compact-open topology ?
As $[0,1]^n$ is not a smooth manifold, I don't really know if it is relevant in my case...
Is $F'$, endowed with this topology, metrizable ?
If you have some advices (or even other propositions), thanks in advance !