A More Intuitive Topology
Topology is defined a bit oddly. Mostly,we know what sort of properties we want a "space" to have, but formalizing them to be defined using set theory is a little trickier.
The mixed metaphor I will use will be maps and lenses. It won't be perfect, since it’s hard have a good intuition for “infinitely many.” Also, many intuitions generalize from very nice topologies called "manifolds," and so won't generalize perfectly.
What we really want to do with space is figure out what is close together, what an individual area looks like, but we don't want to see individual points.
So, a topology should contain sets of points, and think of those as the areas we want.
Now, it can't just be any collection of sets, so let's think of some constraints we can impose on them.
1) If you can see two different areas, then you should be able to see what happens when you put both together.
In set theoretic terms, this means we want it to be closed under union.
There is no reason for it to just be a finite number. If we get infinite areas, when you put them all together, there is no reason for them to not also be like a space.
In fact, infinities let us make different shapes that we also want to be able to use.
So, it should be closed under arbitrary (both finite and infinite) union.
2) If you see two different areas, then you should also be able to only see areas where both are. You could think of putting the two filtering lenses on top of each other, or drawing only where the maps agree.
In set theory terms, we want it to be closed under intersection.
Unlike union, we don't want it to allow all infinities. Think of if you have circles getting smaller and smaller.
Then, if you could see the point they all share as an "area," then we can see any individual point too. This is ok for some topologies, but we don't want it to be a condition for all topologies.
That is, it should be closed under finite intersections (but not necessarily infinite ones).
Note that this makes an asymmetry between the operations, which is fun and interesting.
3) It must contain both the empty set and the entire space.
If we want to see areas that aren't near each other, for completeness, their intersection is the empty set.
If we take all of what we define as areas, they should cover the whole space, because we don't want to focus on anything we can't see.