$\DeclareMathOperator\Ball{Ball}$Question: What "well-known" spaces 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 exist, 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 $\operatorname{colim}_k \left\{f\in C(\mathbb{R}^n)\mathrel: \text{$f(x)=0$ if $ \|x\|>k$}\right\}\mathrel{:=} X_k$ in the category of LCS; note, we can refine the topology by instead taking the limit in $\mathrm{Top}$.
Upon identifying each $X_k$ with the subset of space $C(\overline{\Ball(x,k)})$ consisting of function vanishing on $\partial \operatorname{Ball}(x,k)$, 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 continuity of any function therein.
