Just a small (and, really, quite useless) little nugget here: In the definition of a topological space, we require that arbitrary unions and finite intersections of open sets are open. We also need that the whole space and the empty set are also open sets. However, this last condition is actually redundant - the empty union is empty and, in the context of subsets of some set, the empty intersection is the whole set, so that the whole space and the empty set are open is actually implied by the first two properties! Of course, I wouldn't suggest not including the last condition in the definition when teaching topology, just thought it was interesting to point out :)
Isn't the empty intersection still the empty set? You mean to say [tex]\emptyset\cap\emptyset=S[/tex]? (Where S is the whole set)
I think the OP means "intersection over no sets". A point x belongs to an intersection if and only if it belongs to each of the sets over which the intersection is taken. If the intersection is over no sets, there are no conditions to check. The statement "x belongs to the intersection" is then vacuously true.
Exactly. An arbitrary union indexed over some set is the set of points x which have the property that they belong to one of the indexed sets. Hence, the empty union is always empty. The empty intersection is a little more troublesome in the general context. As you say, the condition is vacuously true, so you get issues with Russell's paradox and so on. These are all fixed if in the context of a universe, or if you are talking about subsets of a particular set (e.g. the set of elements in your topological space). http://en.wikipedia.org/wiki/Intersection_(set_theory)#Nullary_intersection