Let M be a (non-compact) smooth manifold and consider the set $C^\infty_c(M)$ of smooth real-valued functions with compact support. We can give this function space several topologies. Here are four:

**Whitney $C^\infty$-topology**(= weak $C^\infty$-topology) $\tau_1$:

For each $\varepsilon$ a real number, $X = (X_1, \dots, X_r)$ a finite choice of vector fields on $M$ and $K \subseteq M$ a compact set, we define: $$ B(\varepsilon, X, K) = \{ f \in C^\infty_c(M) : |X_1 X_2 \cdots X_r f(x)| < \varepsilon \; \forall x \in K\}$$ Then a subbasis of the topology is given by the sets $g + B(\varepsilon, X, K)$.

**Strong $C^\infty$-topology**$\tau_2$:

Similar to the above: $X = (X_1, \dots, X_r)$ a finite choice of vector fields on $M$, but now $\varepsilon: M \to \mathbb{R}_{>0}$ is a positive function, and we set: $$ B(\varepsilon, X) = \{ f \in C^\infty_c(M) : |X_1 X_2 \cdots X_r f(x)| < \varepsilon(x) \; \forall x \in M\}$$ A subbasis of the topology is given by the sets $g + B(\varepsilon, X)$.

- A
**variant of the Strong $C^\infty$-topology**$\tau_3$:

Similar to the preceding topology except that $X = (X_1, X_2, \dots)$ is an infinite tuple of vector fields (indexed by the natural numbers) which is *locally finite* in the following sense: for each $x \in M$, there exists an open neighborhood $U$ of $x$ such that $X_i|_U = 0$ is the zero vector field for all by finitely many indices $i$. Of course which indices have non-zero vector fields may vary from point to point. As before $\varepsilon: M \to \mathbb{R}_{>0}$ is a positive function, and we set:
$$ B(\varepsilon, X) = \{ f \in C^\infty_c(M) : \max_{S \subseteq \mathbb{N}}|X_{s_1} X_{s_2} \cdots X_{s_r} f(x)| < \varepsilon(x) \; \forall x \in M\}$$
where the maximum is computed over all finite (ordered) subsets $S = (s_1, s_2, \dots, s_r) \subseteq \mathbb{N}$.

A subbasis of the topology is given by the sets $g + B(\varepsilon, X)$.

- A "
**plot topology**" $\tau_4$:

A set theoretic map $p: [0,1] \to C^\infty_c(M)$ will be called a *smooth plot* if the adjoint map
$$ \tilde{p}: [0, 1] \times M \to \mathbb{R}$$
is smooth and there exists a compact subset $K \subseteq M$ such that
$$ \tilde{p}(t, x) = \tilde{p}(0, x)$$
is independent of $t \in [0,1]$ for all $x \in M \setminus K$. In other words outside of a compact subset of $M$ the path $\gamma$ is the constant path.

Then we define a topology on $C^\infty_c(M)$ as follows: A set $U \subseteq C^\infty_c(M)$ is open if and only if for all smooth plots $p$ the set $p^{-1}(U) \subseteq [0,1]$ is open.

Here are a few observations about these topologies. First is that there are several equivalent ways to describe these topologies. For example for the strong topology you might instead have a sequence of $\varepsilon_\alpha$ and charts $U_\alpha$ indexed by the same and then a different tuple $X_\alpha$ for that open set. The version I am using avoids choosing charts.

The topologies $\tau_1$ and $\tau_2$ are considered in Hirsch "Differential Topology". All three topologies $\tau_1$, $\tau_2$, and $\tau_3$ are considered and compared in "Manifolds of Differentiable Mappings" by Michor where they are called the $CO^\infty$, $WO^\infty$, and $LO^\infty$ topologies respectively.

It is not hard to see that we have containments $$\tau_1 \subseteq \tau_2 \subseteq \tau_3$$ Especially the second containment since the sub-basis for $\tau_2$ is contained in the sub-basis for $\tau_3$.

If $M$ is non-compact then these containments are proper. Given a sub-basis $\{B_\alpha\}$ for a topology $\tau_i$, we get a basis by taking finite intersections of the $B_\alpha$. If $B$ is an open set of the next topology $\tau_{i+1}$ which is also open in $\tau_i$, then $B$ has to contain some element of the basis. But in each case it is clear that there are elements of the next sub-basis which do not contain any such finite intersection.

For example if $B(\varepsilon, X)$ is one of the sets defining $\tau_3$ such that $X$ is locally finite, but the number of non-zero vector fields is unbounded in $M$, then $B(\varepsilon, X)$ does not contain any finite intersection of the sub-basic open sets for $\tau_2$. Such finite intersections can only restrict the size of derivatives up to a finite order, while $B(\varepsilon, X)$ restricts the size of arbitrarily large derivatives, but only as we get closer to the "edge" of $M$.

So $\tau_2 \subseteq \tau_3$ is a proper inclusion for $M$ non-compact.

In Michor "Manifolds of Smooth Maps" he also looks at the topology $\tau_3$ (there it is called $\mathcal{D}^\infty$). There he shows (Cor. 2.3) that a sequence of functions $f_n$ converges to $f$ in this topology if and only if there is a compact $K \subseteq M$ such that $f_i = f_j$ outside of $K$ and such that all derivatives of the $f_n$ at all orders converge uniformly to $f$.

From this it follows that a continuous path $[0,1] \to C^\infty_c(M)$ has to consist of a path of functions which is constant outside some compact $K \subseteq M$, just as in the plots used for $\tau_4$. In particular those plots are continuous and so by formal reasons we have $\tau_3 \subseteq \tau_4$.

By a smooth approximation argument you can show that if you used the $\tau_3$-continuous paths instead of the smooth plots I used above, you get the same topology $\tau_4$.

Interestingly $\tau_2$ and $\tau_3$ have the same continuous paths.

Finally all three topologies agree when $M$ is compact. This is classic and not hard for $\tau_1$, $\tau_2$, and $\tau_3$. For $\tau_4$ this is shown in Cor. 4.14 in "The D-topology for Diffeological Spaces" by Christiansen, Sinnamon, Wu.

It it also easy to see that all the topologies agree when you restrict to the subspace $C^\infty_K(M)$ of smooth functions supported on a fixed compact subset. It is only when the compact subsets are allowed to vary that things become interesting.

Also in Kriegl and Michor's book "The Convenient Setting of Global Analysis" I think they also consider (a version of?) topology $\tau_4$. But I am not sure if they compare to topology $\tau_3$.

So my question is:

Question: What is the relationship between the topology $\tau_3$ (a variant of the strong topology) and topology $\tau_4$ (the plot topology). Is the inclusion $\tau_3 \leq \tau_4$ a strict refinement or do they coincide?