# Topologies on compactly supported functions

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:

1. 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 $g + B(\varepsilon, X, K)$.

1. 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 $g + B(\varepsilon, X)$.

1. 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 $g + B(\varepsilon, X)$.

1. 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?

• What do you mean by "the adjoint map"? – Fan Zheng Oct 11 '16 at 16:46
• It is defined as $\tilde{p}(t,x) =p(t)(x)$ (which makes sense since $p(t) \in C^\infty_c(M)$ is itself a function). – Chris Schommer-Pries Oct 11 '16 at 18:58
• I'm not sure if $\tau_3\subseteq\tau_4$ – Fan Zheng Oct 12 '16 at 17:00
• Let's say $M=\mathbb{R}$. Let $X_n=\frac{\rho(nx)-\rho(2nx)}{x^2}\partial_x$, where $\rho$ is a bump function supported on $[-1,1]$ and equals 1 on $[-1/2,1/2]$. Then at $|x|=1/n$, the nonzero vector fields are roughly $X_{n/C},\dots,X_{Cn}$, the maximum coefficient of which is $Cn^2$. Let's just take the open set $U$ with $r=1$ and $\epsilon(x)=1$. Then any $f\in U$ will have $|f'(\pm 1/n)|\le c/n^2$, which forces $f'(0)=0$. Now take the plot $p(x,t)=tx$. Then $p^{-1}(U)=\{0\}$ is not open. – Fan Zheng Oct 12 '16 at 17:08
• You are right, the variant of strong topology is not correct as stated. There are several ways to define this topology, and I was trying to make a definition which avoided choosing open covers. The locally finite condition should be modified to say that around every point there is an open set such that only finitely many of the $X_i$ are non-zero on this open set. That rules out your choice of $X_n$. I will correct the definition of the topology. – Chris Schommer-Pries Oct 12 '16 at 17:21