Consider the space $\mathcal D(\mathbb{R}^n)$ of smooth functions (in the sense of having continuous derivatives of all orders) which are compactly supported. Endow it with its usual topology, i.e., the topology such that the dual space is the space of distributions.


Question: Is $\mathcal D(\mathbb{R}^n)$ separable?



Is the space of test functions separable?
I suppose, you mean the usual topology on $D({\mathbb R}^n)$ defined for example in [Rudin's book](http://rads.stackoverflow.com/amzn/click/0070542368). Take $D\_N=\{\varphi\in D({\mathbb R}^n):\ {\rm supp}\varphi\subseteq\{x\in{\mathbb R}^n:\ |x|\le N\} \}$, where $N\in{\mathbb N}$. Each $D\_N$ can be considered as a subspace of the space $C^\infty({\mathbb R}^n)$ with the induced topology. Since this topology on $C^\infty({\mathbb R}^n)$ has a countable base, $D\_N$ also has a countable base. As a corollary, $D\_N$ is separable. Let $S\_N$ be a countable dense subset in $D\_N$. Then $S=\bigcup\_{N\in{\mathbb N}}S\_N$ is countable and dense in $D({\mathbb R}^n)=\bigcup\_{N\in{\mathbb N}}D\_N$.


