Let $\Omega\subset\mathbb{R}^n$ be open, $\mathscr{C}(\Omega,\mathbb{R})$ the Fréchet space of real-valued continuous functions on $\Omega$ endowed with the compact-open topology, and $\mathscr{C}\_u(\Omega,\mathbb{R})\subset\mathscr{C}(\Omega,\mathbb{R})$ the linear subspace of uniformly continuous real-valued functions on $\Omega$. 


Is $\mathscr{C}\_u(\Omega,\mathbb{R})$ some member of the Borel hierarchy of subsets of $\mathscr{C}(\Omega,\mathbb{R})$? For instance, is $\mathscr{C}\_u(\Omega,\mathbb{R})$
a $G\_\delta$ set, or a $F\_\sigma$ set, in the compact-open topology of $\mathscr{C}(\Omega,\mathbb{R})$?



Uniformly continuous functions and Borel hierarchy in the compact-open topology
Let $(K\_n)\_{n\in\mathbb N}$ be a compact exhaustion of $\Omega$ (that is, every compact set is contained in some $K\_n$) and define
$$ A\_{n,m,k}=\lbrace f\in \mathscr C(\Omega,\mathbb R): \sup\lbrace |f(x)-f(y)|: x,y \in K\_n, d(x,y)<1/m\rbrace < 1/k\rbrace.$$
This set is open with respect to the semi-norm $\|f\|\_n=\sup\lbrace |f(x)|:x\in K\_n\rbrace$
which easily follows from the triangle inequality. Hence
$$ \mathscr C\_u(\Omega,\mathbb R)= \bigcap\_{k\in\mathbb N} \bigcup\_{m\in\mathbb N} \bigcap\_{n\in\mathbb N} A\_{n,m,k}$$ is (at least) $G\_{\delta \sigma \delta}$.


