So I have tried to understand the basics of topology, but I have some trouble getting a good intuition for it. I know that the idea is supposed to be that we have various open sets telling us something about the "nearness" of the points in the space, and together the set of all open sets (the topology) tells us something about the structure of the whole space.
These concepts make a lot of sense to me for metric spaces themselves, where open sets are finite or infinite unions of open balls with various radii. In particular, if we have some point y and some ball centered at it, then this ball gives the notion of a "neighborhood" around y, and the smaller the ball is, the "closer" the points in it will be to y. I thought a lot about the notion of neighbourhoods and open sets for general spaces, and I concluded that, roughly, the same applies to all spaces: if you have some open set in the basis of the topology (like the open balls are in the case of metric spaces) containing a point y, then the "smaller" the set is (in other words, the smaller the number of elements is), the "closer" the points in the set can be imagined to be to y. This then says something about how open sets tell us about the "nearness" of points, as I mentioned in the first paragraph.
But then I had some trouble applying these ideas to find the topologies of even very simple sets containing a small number of elements, which shows that there is something wrong with my intuition above. So, then, what is a better way of thinking about topologies and how they contain information about the "nearness" of points in a set?