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 nonempty set X and choose a nonempty 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 nonempty 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 OpensetMotivation 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