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 $\mathscr{C}(\Omega,\mathbb{R})$?