I am interested in how to prove that the space of smooth differential forms with compact support on some open $U \subset \mathbb{R}^n$, denoted $\mathcal{D}^n(U)$, is separable, i.e. it has a countable dense subset. The topology on $\mathcal{D}^n(U)$ is the canonical LF-topology, the usual locally convex topology endowed on the space of test functions to make its dual space the space of distributions. This problem stems from trying to understand the proof of the Compactness Theorem of geometric measure theory.
If $\{T_j\}\subset \mathcal{D}_m(U)$ is a sequence of integer multiplicity $m$-currents with $$\sup_{j \geq 1}\left( \mathbb{M}_W(T_j)+ \mathbb{M}_W( \partial T_j)\right)< \infty$$ for each $W \subset \subset U$, then there is a subsequence $\{T_{j'}\}$ that weakly converges to some integer multiplicity $T \in \mathcal{D}_m(U)$ in $U$.
Note that $\mathcal{D}_m(U)$ is the space of $m$-currents, and is the continuous dual space of the aforementioned $\mathcal{D}^m(U)$, the space of smooth differential $m$-forms with compact support in $U$, which are in a sense the "distributions" and the differential forms the test functions. $W \subset \subset U$ means that the closure of $W$ is compact and contained within $U$, and the mass of the current $T_j$ on $W$ is defined to be $$\mathbb{M}_W(T_j):=\sup_{\omega \leq 1, \operatorname{supp} \omega \subset W}T_j(\omega)$$ for $\omega \in \mathcal{D}^n(U)$. For the purposes of this question, the integer multiplicity part may be ignored. The way the existence of the subsequence and limit $T$ is established within most GMT literature is via the Banach-Alaoglu Theorem, which states that
Let $X$ be a topological vector space. Then the closed unit ball $\overline{\mathbb{B}}$ of the continuous dual space $X^\ast$ is compact with respect to the weak*-topology, where $$\overline{\mathbb{B}} :=\{\Lambda \in V^\ast: |\Lambda \textbf{x}| \leq 1, \textbf{x} \in V\}.$$
I understand that for the case where $X$ is separable, then the closed unit ball of $X^\ast$ is metrizable and hence compactness of the ball is equivalent to sequential compactness. Hence, if $\mathcal{D}^m(U)$ is separable, I will be able to use the continuous dilation map $L:\mathcal{D}_m(U) \to \mathcal{D}_m(U)$ on the closed unit ball to find a large enough (sequentially) compact ball that contains the whole sequence of currents $\{T_j\} \subset \mathcal{D}_m(U)$, hence obtaining the existence of a limit $T$ and a subsequence converging thereof.
Every proof of the theorem merely cites the Banach-Alaoglu theorem without further elaboration, which leads me to suspect that they all merely assume the existence of a countable dense subset of $\mathcal{D}^m(U)$. In fact, this is verified on page 157 of Leon Simon's Introduction to Geometric Measure Theory, where he states
[...] in order to complete the proof of $(2)$; by considering a countable dense set of $\omega \in \mathcal{D}^n(U)$ one can of course show that $4.2$ is applicable [...]
Perhaps Stone-Weierstrass or some other theorem may be of use here, but I am unsure myself. Thanks in advance!