My purpose for asking this next question is rather difficult to describe. I have been meditating on a problem for several weeks now, so if my line of thinking doesn't make any sense, by all means please tell me so. This is not a problem for any class. It is for my own curiosity. As a result, I don't have any guidance for how to pursue my goals, and my mathematical intuition is still rather underdeveloped in this instance.
Let be some uncountable set. Let define an arbitrary function.
Let have the discrete topology in the range of and pull back the weakest topology under which is continuous. Call this topology . Next, let have the topology in the range of , and again pull back the weakest topology under which is continuous. Call this new topology . Continue this process ad infinitum.
Is it possible to determine if the sequence of topologies converges? Is it even possible to have a convergent sequence of topologies?
I'm interested in the Subset Sum problem. Looking into it, I've found several interesting theorems about it, but with only a few exceptions, they tend to be combinatorial results on a very small number of sets (typically some finite number of cases, or possibly some countable number, and I have yet to find a theorem that works for any uncountable subset of ). For instance, the proof will begin with sets of n integers with values ranging from 1 to 2n+1 and through combinatorics, indicate the range of subset sums that will always be achieved.
In other posts, I've posted questions about the following function:
At first, I simply wanted to partition in order to see what types of sets functioned similarly. Using your suggestion to map the set to the reals, I was hoping to see if I could determine if that sufficed as a metric for each partition such that sets of integers behaved in a similar fashion depending upon which partition they were in. My partition was flawed, however, as the function was extremely discontinuous in most of my partitions. I briefly thought about finding a topology on each partition, until I discovered that many sets in different partitions mapped to the same image implying that my partition was causing some of the discontinuity, and it was not inherent to the function itself.
So, now I am investigating alternative approaches to understand this problem. My eventual goal is to find a technique under which combinatorial results might be analytically extended to some neighborhood of those results (if such a concept can be well-defined in this context). I use the term neighborhood loosely, as I am not sure that I will be capable of finding a topology for this, and without a topology, I obviously can't have a neighborhood. So, a more formal way to state this would be, I am looking for the largest subset of upon which I can induce a topology (potentially the trivial topology) in order to examine instances where sets behave similarly. If, for a particular problem, my largest subset winds up being precisely the set already proven by the theorem, I will likely conclude that this method cannot produce the machinery I hoped it could.
I apologize if my explanation is vague. I am still at the beginning of my investigations into this problem, and I am finding it difficult to focus my efforts towards explaining my intent when I have so many thoughts bouncing around about possible approaches to the problem.
I hope this lengthy explanation is comprehensible enough to give you the general idea of my purpose. The nature of the problem makes calculations of discrete examples rather limited, so checking specific examples is tedious. Therefore, I am checking examples sparingly, attempting to garnish specific information if/when I do check specific examples. For instance, that is how I was able to eventually find that an uncountable number of sets of integers had multiple preimages, and later that my partitioning sets based on their number of preimages was not terribly helpful.
So, back to your question of whether I have any conjectures about this: no. At the moment, I unfortunately do not. I have an ideal for what I hope to be true. I hope that I can topologize all of , and I hope that topology is metrizable. These lofty goals seem unlikely, given the nature of the problem. And I have a variety of lesser goals ranging from abandoning this method of inquiry entirely in favor of more concrete investigation until I understand the problem better, or possibly abstracting the concept further by looking at the set of all functions and trying to determine some underlying properties before investigating the specific case of the one function that currently holds my interest.
Thinking about this more, I realized that I am, indeed, looking for if the sequence of topologies converges. I am looking for them to converge in with the discrete topology.
I have not yet proven this, but it seems obvious to me that should a limit exist, it will have this property: If , and if in the discrete topology then every neighborhood containing also contains and visa versa. This would imply that the limit of the sequence of topologies (again, should it exist) is not Hausdorff, or even . Therefore, it is certainly not metrizable.
I am still thinking about how to show if this limit of topologies exists.