UPDATE (27/08/2020): I realized after a comment from Jochen Wengenroth that there was at least one false premise behind my question, owing to the fact that analysts sometimes use the words "inductive limit topology" to mean something different than a topologist would. I have now edited the question with this in mind, and put in an additional question at the end.

My question is similar to this question about relating different topologies on $C_c^\infty(M)$, but I'm wondering about a few specific issues that weren't cleared up by that question. The short version of my question is:

Which of the commonly used "strong" topologies on the space of smooth compactly supported functions are equivalent to each other?

I have developed a definite opinion on what the answer should be, but it conflicts in part with things I've read elsewhere, including in Hirsch's Differential Topology book -- so if Hirsch is right and I am wrong, I need someone to tell me why.

Concretely, fix an open set $\Omega \subset {\mathbb R}^n$ and consider the space $C_c^\infty(\Omega)$ of smooth real-valued functions with compact support. (We could also talk about functions on a smooth manifold, or maps from one manifold to another, but let's not make this more complicated than it needs to be.) Borrowing some terminology from Daniel Bruegmann's excellent answer to the other question, I would like to compare the following topologies on $C_c^\infty(\Omega)$:

  1. The Whitney $C^\infty$-topology: ${\mathcal U} \subset C_c^\infty(\Omega)$ open means that for every $\varphi \in {\mathcal U}$, there exists an integer $k \ge 0$ and a collection of continuous functions $f_\alpha : \Omega \to (0,\infty)$ such that every $\psi \in C_c^\infty(\Omega)$ with $|\partial^\alpha(\psi - \varphi)| < f_\alpha$ for all multi-indices $\alpha$ of order at most $k$ belongs to ${\mathcal U}$. Equivalently, this is the coarsest topology such that for every $k \ge 0$, the $k$-jet map $$ j^k : C_c^\infty(\Omega) \to C^0(\Omega,J^k(\Omega,{\mathbb R})) $$ is continuous, with the space $C^0(\Omega,J^k(\Omega,{\mathbb R}))$ of continuous sections of the $k$-jet bundle endowed with the strong Whitney $C^0$-topology.
  2. The strong $C^\infty$-topology: a neighborhood base of $\varphi \in C_c^\infty(\Omega)$ is given by all sets of the form $$ \left\{ \psi \in C_c^\infty(\Omega)\ \Big|\ \|\psi - \varphi\|_{C^{k_i}(\Omega_i)} < \epsilon_i \text{ for all $i \in I$} \right\} $$ where $\{\Omega_i\}_{i \in I}$ is an arbitrary locally finite open covering of $\Omega$, and $\{k_i\}_{i \in I}$ and $\{\epsilon_i\}_{i \in I}$ are arbitrary collections of nonnegative integers and positive real numbers respectively. Unless I'm mistaken, this is the same as what has sometimes been called the very strong $C^\infty$-topology, e.g. in this paper, in order to contrast it with 1 (which is also sometimes called the "strong $C^\infty$-topology").
  3. This one is equivalent to either 1 or 2, depending on whom you ask: the coarsest topology such that the infinity-jet map $$ j^\infty : C_c^\infty(\Omega) \to C^0(\Omega,J^\infty(\Omega,{\mathbb R})) $$ is continuous, where $C^0(\Omega,J^\infty(\Omega,{\mathbb R}))$ is endowed with the strong $C^0$-topology and $J^\infty(\Omega,{\mathbb R})$ with the inverse limit topology with respect to the sequence of natural projections $\Omega \times {\mathbb R} = J^0(\Omega,{\mathbb R}) \leftarrow J^1(\Omega,{\mathbb R}) \leftarrow J^2(\Omega,{\mathbb R}) \leftarrow \ldots \leftarrow J^\infty(\Omega,{\mathbb R})$.
  4. The locally convex inductive limit topology: endowing $$C_K^\infty(\Omega) := \{ \varphi \in C_c^\infty(\Omega)\ |\ \text{supp}(\varphi) \subset K \} $$ for each compact subset $K \subset \Omega$ with its natural Fréchet $C^\infty$-topology, one endows $C_c^\infty(\Omega)$ with the finest locally convex topology for which the inclusions $C_K^\infty(\Omega) \hookrightarrow C_c^\infty(\Omega)$ are all continuous. In other words, the topology of $C_c^\infty(\Omega)$ is generated by the collection of all seminorms whose restrictions to $C_K^\infty(\Omega)$ for all compact $K \subset \Omega$ are continuous. (I borrowed this version of the definition from a very nice blog post by Terry Tao.) This topology does not seem to be mentioned often by differential topologists, but is well known to analysts as the topology of the space of test functions in the theory of distributions.
  5. The direct limit topology: the finest (not necessarily locally convex) topology such that the inclusions $C_K^\infty(\Omega) \hookrightarrow C_c^\infty(\Omega)$ are continuous (with respect to the natural Fréchet space topology on $C_K^\infty(\Omega)$) for all $K \subset \Omega$ compact. In other words, ${\mathcal U} \subset C_c^\infty(\Omega)$ is open if and only if ${\mathcal U} \cap C_K^\infty(\Omega)$ is an open subset of $C_K^\infty(\Omega)$ for every $K \subset \Omega$ compact. (This was 4 in the original version of the question, but since the space of test functions was what I actually meant to talk about, I've changed this one to 5 in the edit.)

Question 1: Are 1 and 3 equivalent? This impression emerges from Hirsch's book, which defines the strong topology for $C^\infty$ maps between two manifolds as 1 on page 36, and then on page 62 casually states without proof that it is equivalent to 3. Perhaps I am misunderstanding what Hirsch intended to say, but I believe this is wrong.

Here are some things that I believe to be true, and I shall be very grateful if anyone can confirm or refute them:

  • If the definition of 2 were modified to require that the families $\{k_i\}_{i \in I}$ are always bounded, then it would become equivalent to 1. As written, however, 2 is a strictly finer topology than 1, even though they have the same notion of convergent sequences (which must always have support in a fixed compact subset).
  • Topologies 2 and 3 are equivalent. Appendix C of this paper appears to give a proof of this. The tricky part is to understand what exactly the inverse limit topology on $J^\infty(\Omega,{\mathbb R})$ is; I think the key point is that an open subset of $J^\infty(\Omega,{\mathbb R})$ constrains only finitely many derivatives over sufficiently small neighborhoods of any point in $\Omega$, and the same is therefore true over any compact subset $K \subset \Omega$, but it may still constrain derivatives of unboundedly high order as one moves out toward infinity.
  • Topology 4 is also equivalent to 2 (and therefore 3).

Actually, let's reformulate that as Question 2: is 4 equivalent to 2 and/or 3?

It seems almost obvious if one stares long enough at the family of seminorms for 4 given in this answer to another question. Nonetheless, I have been struggling to find a clear and unambiguous statement of this equivalence, and my suspicion is that the only reason it's so hard to find is that topologists and analysts do not always talk to each other as much as they should. The rest of my confusion is probably due to the fact that the words "strong $C^\infty$-topology" are often used in the literature without clearly specifying whether 1 or 2 is meant -- indeed, it had not occurred to me on first glance that they are different. But I have in the mean time convinced myself that there exist distributions $C_c^\infty({\mathbb R}^n) \to {\mathbb R}$ that are not continuous with respect to 1, so if I'm correct about 4 and 2 being equivalent, this proves 2 to be strictly finer than 1.

Added in the edit (27/08/2020): Topology 5 is very natural from an abstract perspective, and appears to have the same notion of convergent sequences as all the others. It also has the very nice property that for any topological space $Y$, a map $f : C_c^\infty(\Omega) \to Y$ is continuous if and only if it is sequentially continuous. (For topology 4, I only know how to prove that when $f$ is linear.) But this topology doesn't appear to be used in analysis, and I assume the reason is that it is not locally convex, so it doesn't have the nice properties that locally convex spaces have. But is that obvious?

Question 3: Is 5 strictly finer than 4, or equivalently, is 5 really not locally convex?

I can see where one runs into a roadblock in trying to prove that 5 is contained in either 2 or 4, and I've made some effort to construct a counterexample, but can't quite see it. Is there, for instance, a (necessarily nonlinear) map $f : C_c^\infty(\Omega) \to Y$ that is sequentially continuous (and therefore continuous with respect to 5) but not continuous with respect to 4?

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    $\begingroup$ I guess that in 4 you mean the locally convex inductive limit, and I think that the statement only holds for convex sets $\mathcal U$. $\endgroup$ – Jochen Wengenroth Aug 25 '20 at 17:55
  • $\begingroup$ Thought quickly, it seems to me that $1=3<2=4$ provided that Jochen's comment is taken into account. $\endgroup$ – TaQ Aug 26 '20 at 0:42
  • $\begingroup$ @Jochen, many thanks for your comment, which has just given rise to a major edit of the question, as I realized I had misunderstood something. $\endgroup$ – Chris Wendl Aug 27 '20 at 14:28
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    $\begingroup$ After a more thorough thought, and having looked at Lemma 41.11 on page 437 in Kriegl and Michor's book The Convenient Setting of Global Analysis, it seems that $1<2=3=4<5$. For $4<5$ the argument is that the Mackey closure topology / $c^\infty$ topology for $C_c^\infty(\Omega)$ makes the embeddings $C_K^\infty(\Omega)\hookrightarrow C_c^\infty(\Omega)$ continuous and is not even a vector space topology by Proposition 6.2.8(ii) on page 195 in Frölicher and Kriegl's Linear Spaces and Differentiation Theory. $\endgroup$ – TaQ Aug 27 '20 at 17:44
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    $\begingroup$ If I remember correctly, all these topologies were treated in chapter 4 of my book Manifolds of Differentiable Mappings (1980), (mat.univie.ac.at/~michor/…). It is shown there that the two topologies mentioned in the book of Hirsch differ. See also the paper: Neves, Vítor: Nonnormality of 𝒞∞(M,N) in Whitney's and related topologies when M is open. Topology Appl. 39 (1991), no. 2, 113–122. $\endgroup$ – Peter Michor Sep 1 '20 at 19:19

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