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**SlipEternal** 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 $\displaystyle X$ be some uncountable set. Let $\displaystyle f: X \to X$ define an arbitrary function.

Let $\displaystyle X$ have the discrete topology in the range of $\displaystyle f$ and pull back the weakest topology under which $\displaystyle f$ is continuous. Call this topology $\displaystyle \mathcal{T}_1$. Next, let $\displaystyle X$ have the topology $\displaystyle \mathcal{T}_1$ in the range of $\displaystyle f$, and again pull back the weakest topology under which $\displaystyle f$ is continuous. Call this new topology $\displaystyle \mathcal{T}_2$. Continue this process ad infinitum.

Is it possible to determine if the sequence of topologies $\displaystyle \mathcal{T}_n$ converges? Is it even possible to have a convergent sequence of topologies?