ah, so $f^{\leftarrow}(\{0\})=X$ is closed by continuity of $f$
and thus we have $\overline{X}=X$ in this case
Consider the following topology (NB have not drew all the open sets as otherwise the whole figure will become very cluttered, it is to be understood that all open sets are given by finite intersections and unions of what is being drawn here)
What I noticed is that the more number of open sets a point is being enclosed in, the "harder" it is to converge to if it is a limit point since there are less possible nets that can have said limit
So the point at the furthest left will involve mostly nets that are eventually constant
whereas points that will converge to the topmost point, said net can have more variations in the middle
This also seemed to fit quite well with the intuition on large vs small topology
So the trivial topology is one where everything converges to everything, and the discrete topology is one where the only convergent nets are the eventually constant ones
I wonder, whether there is a notion of "difficulty of convergence" can be defined for each point by the cardinality of all nets that converge to set point (probably not useful if the underlying set is infinite, because then every such set of convergent nets will be of infinite cardinality for each point)
But can some generalisation of topology forgets even the details of the limit points, and instead can only answer the question of "Given a set X, is its closure itself" with yes/no (meaning that the set of limit points is either empty or nonempty)
The purpose of wanting this kind of generalisation is it might provide me a framework for exploring this philosophical thing:
in The Symposium, yesterday, by
Secret The destination of the Path of The Unknown cannot be planned in advance, nor it is possible to plot, but the destination is reachable in principle, as what used to be part of the unknown, with understanding, becomes part of the known
Which to give a simple example on what this means: Recall how when one is engaging in artistic activity, one does not always have an idea what the product looks like, but there is always going to be a product regardless on what happens in the process of making the artwork
Thus in a sense, the outcome of the artistic activity is unknown, until the artwork is finished. The best we can say is that the product exists, but we cannot specify further what the product will be
I think in order to get a formalisation of this idea, I will need to apply some forgetful functor on the category Top such that it forgets the details of the limit points, and instead reduce that into a question on whether limit points exists given a set
