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I am wondering how a topology can be defined on a set when all that is given is a sense of convergence.

An example is $C^\infty_c$, the space of infinitely differentiable functions with compact support. A sequence $\{u_i\}$ is said to converge to $u$ in $C_c^\infty$ if all the $u_i$'s are supported on a common compact set and $$\|\partial^\alpha (u_i - u)\|_\infty \rightarrow 0$$ for all multiindices $\alpha$. I saw this definition on Folland's text on PDEs. He does not explicitly define $\|\cdot\|_\infty$ but I assume he means the infinity norm viewed as a maximum and that the above is to be interpreted uniformly. Now this notion of convergence somehow defines a topology on $C_c^\infty$, but this is not immediately clear to me. What do the open sets look like? Further, how would one discuss continuity using solely this notion of convergence? Alternatively, Folland says that $C_0^\infty$ is a Frechet space under the family of norms $\|\partial^\alpha u\|_\infty$ and we put the strict inductive limit topology on $C_0^\infty$, however I am unfamiliar with the strict inductive limit topology.

Note I took a look at this question Topology induced by a convergence notion but the explanation is still unclear to me.

CBBAM
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  • The topology of $C_c^{\infty}$ is defined by a sequence of semi-norms. Rudin's FA book has details. – geetha290krm Sep 08 '22 at 23:32
  • @geetha290krm Thank you I will take a look. So the topology is defined by the seminorm, and that topology then gives us a notion of continuity, but there is no direct connection between continuity and the seminorm similar to an $\epsilon$-$\delta$ condition? – CBBAM Sep 08 '22 at 23:37
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    @CBBAM Folland himself says that he will not define the topology on $C_c^{\infty}$, but rather just give the condition for sequences to converge. For Hausdorff topological spaces whose topology is defined with seminorms, there are conditions for continuity of a linear map analogous for the condition for the Banach space case. Folland does cover these spaces in some detail. – Mason Sep 09 '22 at 00:20
  • A topology is not determined by its convergent sequences. You need the convergence of nets. – Jochen Sep 11 '22 at 13:43
  • For a description of the topology of $C_{c}^{\infty}$ see https://math.stackexchange.com/questions/3510982/doubt-in-understanding-space-mathscr-d-omega/3511753#3511753 – Abdelmalek Abdesselam Sep 13 '22 at 18:37

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A topology is defined by somehow specifying which sets are open. One way to do this is to specify which sets are closed instead, as a set is open if and only if its complement is closed.

Suppose you have some concept of convergence of sequences in your space. Presumably, this concept has the property that any subsequence of a convergent sequence is also convergent, and to the same limit. Otherwise, you don't have an adequate concept of convergence.

You can define a topology on your space by saying a set $C$ is closed if and only if it contains the limits of all convergent sequences contained in it. Clearly the empty set and entire space are closed under this definition. If $C$ and $D$ are closed, then any convergent sequence in $C \cup D$ must have an infinite subsequence in one of them. That subsequence converges, and its limit must be in the same space. But that is also the limit of the original sequence, which therefore must be within $C \cup D$. Thus $C\cup D$ is closed. And finally any intersection of closed sets is closed, because any convergent sequences in the intersection are also in each of the closed sets, therefore so is its limit.

So this does define a topology. But there is a caveat. From a topology, you can define the concept of convergence of sequences. But that concept may not match the original concept you had for convergence of sequences. All your original convergent sequences will continue to converge to the same limits under this topology. But it is possible they may converge to other limits as well (i.e., the topology need not be T1). And other sequences that were not convergent in your original concept may converge in the topological sense.

This is why convergence of sequences is not a preferred method for defining topologies.

Paul Sinclair
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  • Thank you this was very helpful! So as a follow up, is it possible to define a topology solely in terms of convergence that is consistent with the topological notion of convergence or that has other properties (e.g. Hausdorffness). – CBBAM Sep 09 '22 at 20:44
  • That is going to depend on what properties your original concept had. I'm not an expert on this matter. I know the basics, but have never looked deeper. So I cannot say what the conditions are that will allow you to get the topological properties you want. As I said, this is not a popular method of defining topologies, because it is hard to control the result. – Paul Sinclair Sep 09 '22 at 21:04
  • For most propbability spaces, almost sure convergence is not the convergence with respect to a topology because the latter notion has the property that a sequence converges to a limit $x$ if and only if every subsequence has a further subsequence converging to $x$. Recall that convergence in measure (which is weaker than a.s. convergence and in most cases strictly weaker) is characterized by the property that every subsequence has another subsequence converging almost surely. This can be found in many books on measure theory. – Jochen Sep 11 '22 at 13:42
  • @Jochen - Convergence with respect to a topology means that every subsequence of a convergent sequence converges to the same limit. There is no need to take a "further subsequence". If there is even one subsequence that fails to converge to the same limit, the full sequence cannot converge. Beyond that, I fail to see what point you are trying to make. – Paul Sinclair Sep 11 '22 at 14:09
  • The point is that almost sure convergence of sequences is not the convergence with respect to any topology. Note that every subsequence of an a.s. convergent sequence converges a.s. to the same limit. – Jochen Sep 12 '22 at 14:31
  • The property Every subsequence of a convergent sequence converges to the same limit should (and does) hold for every sensible notion of convergent sequence (as far as I remeber, something like this is already in M.Fréchet's thesis). The property A sequence converges to $x$ if and only if every subsequence has a further subsequence converging to $x$ is much stronger and does hold for convergence in topological spaces but not for a.s. convergence. – Jochen Sep 12 '22 at 15:04
  • Sorry - I recognized the significance of the futher subsequence after posting my previous reply and deleted it, but obviously you saw it and were replying already yourself. So you are saying that in some probability spaces, there exists a sequence and some value $A$ such that such that every subsequence has itself a further subsequence which converges a.s. to $A$, but the sequence itself does not converge a.s. to $A$. Thus convergence a.s. is an example of a sequence convergence concept from which the topology defined as in my post does not give the same concept of convergence of a sequence? – Paul Sinclair Sep 12 '22 at 15:07