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?