Is the projective limit $\mathcal{D}(\mathbb{R})$ separable?

Question

Let $\mathscr{D}(\mathbb{R})$ be the set $C_0^\infty(\mathbb{R})$ of smooth functions with compact support endowed with the following topology:

The initial topology with respect to the family maps $(\iota_\tau)_{\tau \in T}$, when $\tau=(\tau_1,\tau_2(x))\in T$ and $\tau_1 \in \mathbb{N}$, $\tau_2(x)\in C^\infty(\mathbb{R})$ with $\tau_2\ge1$. And

$$\iota_\tau: C_0^\infty(\mathbb{R})\longrightarrow W^{\tau_1}_2(\mathbb{R},\tau_2(x)dx)$$

is the natural inclusion into given weighted Sobolev space. A family of neighbourhood base is therefore $$\lbrace \psi \in C_0^\infty(\mathbb{R}) : \|\psi-\varphi\|_{W^{\tau_1}_2(\mathbb{R},\tau_2(x)dx)}<\varepsilon\rbrace\qquad (\varphi \in C_0^\infty(\mathbb{R}),\tau \in T, \varepsilon>0).$$

This space is a locally convex space with convergence similar to what one might call classical convergence of test functions. I am asking myself the question if this space is separable and how to prove it.

Answer 1

The projective topology topology you describe is coarser than the usual inductive limit topology on $\mathscr D(\mathbb R)$ (the universal properties imply that it is enough to have continuity of the inclusion of the Fréchet space $\mathscr D([-n,n])$ of smooth function with support in the interval into $W_2^{\tau_1}(\mathbb R,\tau_2(x)dx$).

$\mathscr D(\mathbb R)$ with the finer topology is separable because so are all $\mathscr D([-n,n])$ and hence it is also separable for the coarser topology.