The sequential topology on NNN is not regular

The compact-open topology on the set of continuous functionals from the Baire space to the natural numbers is well known to be zero-dimensional. We prove that the closely related sequential topology on this set is not even regular. The sequential topology arises naturally as the topology carried by the exponential N((NN)) formed in various cartesian closed categories of topological spaces. Moreover, we give an example of an effectively open subset of N((NN)) that violates regularity. The topological properties of N((NN)) are known to be closely related to an open problem in Computable Analysis. We also show that the sequential topology on the space of continuous real-valued functions on a Polish space need not be regular.