Why must a topology on a set contain the empty set? I have just had my first week of topology, and I have a question that is rather basic.
Why must the empty set be an element of any given topology?

(For reference, the definition of a topology T I am working with, for a set X:
1. X and the empty set must be in T.
2. the union of elements of any subset in T, is also in T.
3. the intersection of elements of any finite subcollection in T, is also in T.)

Two important concepts in topology are interior and closure of a set $A\subseteq X$. (Interior is the largest open subset, closure is the smallest closed super-set.) If $\emptyset$ would not be open, then interiors and closures would be undefined in general. Indeed, if the full space $X$ is not closed (in itself), what would be the closure $\overline{X}$?
