$\DeclareMathOperator\Ball{Ball}$**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](https://mathoverflow.net/questions/351605/l-infty-as-colimit#comment881397_351609) in [this post][2].)


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**Background/Motivation:**
The question is motivated by [this failed post][1].  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.  



  [1]: https://mathoverflow.net/questions/376293/density-of-functions-into-the-circle-glueing/376300#376300
  [2]: https://mathoverflow.net/questions/351605/l-infty-as-colimit