In many cases we can see statements like "view $\theta$ in the function $f(\theta, x)$ as parameters and $f$ as function $f_{\theta}$ on $x$". Intuitively we guess if $f$ is continuous the $f_{\theta}$ "continuously depends on $\theta$".
More formally, we start from 2 topological space, parameter space and variable space, viz $\Theta$ and $X$, and we get two set $M=\{\mu:\Theta\times X\to \mathbb{R}|\mu\ is\ continuous\} $,$R=\{\rho :X\to\mathbb{R}|\rho\ is \ continuous\}$, where $\Theta\times X$ is equipped with product topology. What we say above now is exhibited in the explicit way that we wish all the maps as $\bar{\mu}:\Theta\to R,\ \bar{\mu}_{\theta}(x)\equiv \mu(\theta,x)$ are continuous, and of course this condition requires a finest topology on the map space $R=\{\rho :X\to\mathbb{R}|\rho\ is \ continuous\}$.
Then my question is that is this finest topology unique in some sense? And if so, what is it exactly? In particular, I am interested in the ordinary case $\Theta=\mathbb{R}^{n}$, $X=\mathbb{R}^{m}$, what will this topological space $R$ be then?
Thanks a lot.
