Question: Are there any other "well-known" spaces which can be explicitly written down in the form $\bigcup_k \phi_k C(K_n,\mathbb{R}^m)$; where $K_n$ is a non-empty compact subset of some Euclidean space, $\bigcup_{k} \phi_k C(K_n,\mathbb{R}^m)$ is equipped with a topology which is no finer than the colimit topology, and most importantly, each $\phi_k$ is explicitly given.

Note: I know that such $\phi_k$ typically exists but I really only care about examples where they are known explicitly. (See Bill Johnson's comment in this post).

Background/Motivation: The question is motivated by this failed post. It is well-known that the space $C_c(\mathbb{R}^n)$ of compactly-supported functions with fine topology can be expressed as the colimit $\operatorname{colim}_k \left\{f\in C(\mathbb{R}^n):\, f(x)=0 \mbox{ if } \|x\|>k\right\}:= X_k$ in the category of LCS; note, we can refine the topology by instead taking the limit in $Top$.

Upon identifying each $X_k$ with the space $C(\overline{Ball(x,k)})$, in the obvious way by "extension by zero", we get the explicit representation of $C_c(\mathbb{R}^n)$ with this topology as $\bigcup_k \iota_{k}\left[C(\overline{Ball(x,k)})\right]$; where $\iota_k$ is the extension by zero of any function in $C(\overline{Ball(x,k)})$. Note, this is well-defined by the continuitiy of any function therein.