It might be a futile attempt to add anything worthwhile to this long list of interesting answers, but let me add my own pedestrian $0.02:

One may think of topology as a set of rules about what's close to what. In other words, it tells me that if I pick a point in the space, then there are several rules (i.e., open sets) that tell me that with respect to some question this set of points is "close" to my chosen point. Considering many rules (i.e., intersecting the open sets) gives me better and better approximation of which points are "really close" to the chosen one. It seems clear that then the union and intersection of these rules would have to belong to the rules.

If we were in a Euclidean space, then we might agree that one way to measure what's close is to put a small (open) ball around a point. If we can't measure, we can't do this, so we need to do something more general and a single open set will not be enough (it's not enough even in a Euclidean space as the radius of the ball that defines "closeness" would certainly depend on the way we want to measure closeness).

So far both open and closed sets would qualify for these rules, but I feel that open sets work better: A rule of "closeness" should be independent of any single point. In other words, a rule should behave the same with respect to any point it applies to (i.e., any point contained in the corresponding set). This clearly picks open sets over close sets.

I suppose one might say that none of this explains what happens with infinitely many rules/sets. I suppose we could say that if we take an infinite set of rules that define closeness, then on one hand we might still say that satisfying any one of the rules is still a reasonable rule while satisfying all the rules is a little bit too much to ask. If you feel this part of my argument is a little shaky, then we agree. I don't have a very good explanation for the behavior of infinite unions and intersections. If I was indeed trying to explain this to undergrads, then at this point I would probably switch over to see what happens in a Euclidean space with all this non-sense about rules of "closeness" and come to the conclusion that a good way to define rules is to say that their corresponding sets contain little balls around every point in them. Then deduce the axioms of open sets in a topology and then say that we should see what these give us if we forget that we were in a Euclidean space.

It might be a futile attempt to add anything worthwhile to this long list of interesting answers, but let me add my own pedestrian $0.02:

One may think of topology as a set of rules about what's close to what. In other words, it tells me that if I pick a point in the space, then there are several rules (i.e., open sets) that tell me that with respect to some question this set of points is "close" to my chosen point. Considering many rules (i.e., intersecting the open sets) gives me better and better approximation of which points are "really close" to the chosen one. It seems clear that then the union and intersection of these rules would have to belong to the rules.

If we were in a Euclidean space, then we might agree that one way to measure what's close is to put a small (open) ball around a point. If we can't measure, we can't do this, so we need to do something more general and a single open set will not be enough (it's not enough even in a Euclidean space as the radius of the ball that defines "closeness" would certainly depend on the way we want to measure closeness).

So far both open and closed sets would qualify for these rules, but I feel that open sets work better: A rule of "closeness" should be independent of any single point. In other words, a rule should behave the same with respect to any point it applies to (i.e., any point contained in the corresponding set). This clearly picks open sets over closed sets.

I suppose one might say that none of this explains what happens with infinitely many rules/sets. I suppose we could say that if we take an infinite set of rules that define closeness, then on one hand we might still say that satisfying any one of the rules is still a reasonable rule while satisfying all the rules is a little bit too much to ask. If you feel this part of my argument is a little shaky, then we agree. I don't have a very good explanation for the behavior of infinite unions and intersections. If I was indeed trying to explain this to undergrads, then at this point I would probably switch over to see what happens in a Euclidean space with all this non-sense about rules of "closeness" and come to the conclusion that a good way to define rules is to say that their corresponding sets contain little balls around every point in them. Then deduce the axioms of open sets in a topology and then say that we should see what these give us if we forget that we were in a Euclidean space.