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?
To each their own, and let a thousand intuitions bloom, but here's one opinion:
I would not adopt the "distance" intuition because everything will go wrong later. For example, uniform continuity is not a topological property, it is a combination of metric and topological ideas; another example is non-Hausdorff spaces, where distinct points can't necessarily be placed in separate neighborhoods. As things get more advanced, you start to realize a lot of important properties and structures depend on metrics that are really the "nearness" concept, and not so much topologies. For example, I am answering this soft question because I am frustrated that I can't get upper hemi-continuity to work the way I want using a metric, because the sequential and topological definitions of uhc are equivalent, but the standard metric definition of uhc implies the topological one but not the converse.
The topological definition of continuity is: ``$f$ is continuous if the inverse image of every open set is open''. The whole point is to get rid of the metric, and think about properties of spaces that are preserved under continuous mappings, like connectedness, compactness, etc. The basic idea of topologies is to remove all the extra structure around the $\varepsilon/\delta$ or sequential definitions of continuity, and replace it with mappings between sets for which the topological definition is the "right" or only one. So if we throw out all of math and start over except for the idea of continuously stretching and shrinking and manipulating sets in continuous ways, which results are clearer and extend to more abstract spaces and what results do we lose? That's topology.