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Did you hear this with relation to an Euclidean topology or some other naturally occuring topology? It cannot be redundant in general since if we take a non-empty set X and choose a non-empty proper subset U of X, then the ""topology"" on X consisting only of U would satisfy the two axioms you stated but would not contain X nor the empty set. Of course, this is a trivial example. 
Suppose we call a collection of subsets of a set X a ""relevant topology"" (this is not a real name; just something I made up just now so it is easier to talk about these things) if it satisfies every axiom except possibly that ""X and the empty set are in the topology"". Then a non-empty set with a ""relevant topology"" that is Hausdorff (here ""Hausdorff"" means that any two distinct points are separated by neighborhoods; the same definition as the usual one on topological spaces) would necessarily have to include the empty set in the ""topology"". Thus for ""many"" topological spaces, requiring the empty set to be open is indeed redundant. 
On the other hand, the requirement that the whole set X be open is in general not redundant with fairly natural topologies (I suppose). For example, take any topological space X and let Y be a set containing X as a proper subset. Declare a subset of Y to be open if and only if it is in the topology on X. Then this ""relevant topology"" on Y satisfies every axiom that a topological space should satisfy except that Y itself is not open. For example, X could be some Euclidean space with the standard topology (and Y could be any set containing X as a proper subset). But this construction is probably not very interesting since (at least as far as I can see) there is not any new information or insight obtained.
This might be too elementary for you, but I wrote the following section in open set a while back that roughly describes why the axioms are the way they are and the motivation for the abstract formulation of a topology: Open_set#Motivation. The following discussion at MathOverflow may perhaps be more appropriate to your question: Do the empty set and the entire set really need to be open?. T "