Below is an extract from Le Gall's Brownian Motion, Martingales, and Stochastic Calculus, p27. I am having trouble seeing why "$\mathscr{C}$ coincides with the Borel $\sigma$-field on $C(\mathbf{R}_+,\mathbf{R})$ associated with the topology of uniform convergence on every compact set". First of all, do we agree that he is referring to the following topology? compact-open topology
Also in the link above, it is said that "If $X$ is compact, and $Y$ is a metric space with metric $d$, then the compact-open topology on $C(X, Y)$ is metrisable, and a metric for it is given by $e( f , g) = \sup\{d( f (x), g(x)) : x \text{ in } X\}$, for $f , g$ in $C(X, Y)$. Does anyone have an idea of why this is or where I might find a proof?
I ask this because if we just consider the Brownian motion of $[0,T]$ instead on $\mathbf{R}_+$, we can equip $C(\mathbf{R}_+,\mathbf{R})$ with the topology associated to the sup norm which is a lot more intuitive to me than what is proposed in Le Gall and seems to be a restriction to $X$ is compact, and $Y$ is a metric space of the second option Le Gall offers.
Thank you in advance!
