My book gives the following as an exercise:

Let (X,T) be any topological space. Verify that the intersection of any finite number of members of T is a member of T. [Hint: to prove this result use "mathematical induction."]

Doesn't this follow directly from the definition of a topology? If this weren't true, then T wouldn't be a topology and (X,T) wouldn't be a topological space. Or so I think. If I'm right, why do we need to prove this at all?

And if I do need to prove it, I'm confused about how to use induction. Let me see ... if u belongs to X, then {u} belongs to T; if u is the only member of X, then {u}, {X} and the null set are the only members of T and it's true. Now if there's a second element v belonging to X, then {v} also belongs to T, so {u} and {v} both belong to T and their intersection (if any) also belongs to T by definition of a topology ... except then I'm assuming what I'm trying to prove, am I not?

I hope one of you guys can straighten me out on this.