Let $\{\omega_i\}_{i\in I}$ be a non-empty set of increasing (not necessarily strictly) continuous functions preserving $0$. Then, for each $i \in I$ define the space $$ C_{\omega_i}(\mathbb{R}^n,\mathbb{R}^d):= \left\{ f \in (\mathbb{R}^n,\mathbb{R}^d):\, \|f\|_{\omega_i,\infty}<\infty \right\} \mbox{ where } \|f\|_{\omega_i,\infty}:= \sup_{x \in \mathbb{R}^n} (\omega_i(\|x\|)+1)^{-1}\|f(x)\|. $$ Then this is a Banach space since the map $f \mapsto f (\omega_i(\|x\|)+1)$ is clearly an isometry with $C_0(\mathbb{R}^n,\mathbb{R}^d)$ for the sup-norm, and the maps between $C_{\omega_i}(\mathbb{R}^n,\mathbb{R}^d)$ and $C_{\omega_j}(\mathbb{R}^n,\mathbb{R}^d)$ can be defined similarly by rescaling analogously. This makes $I$ into a poset with $$ i\leq j \mbox{ if and only if } \sup_{x \in \mathbb{R}^n}\omega_i(\|x\|) \leq \sup_{x \in \mathbb{R}^n}\omega_j(\|x\|). $$ Thus, we can define the LCS colimit of this co-cone $\left\{C_{\omega_i}(\mathbb{R}^n,\mathbb{R}^d)\right\}$.

Now for the question, if $\{\omega_i\}_i$ is taken to be the collection of *all* monotonically increasing and continuous functions identifying $0$ (i.e.: $\omega(0)=0$) then does $\operatorname{co-lim}_i C_{\omega_i}(\mathbb{R}^n,\mathbb{R}^d)$ contain all uniformly continuous functions?

**Note:** Here the colimit is in the LCS sense.