The sequential topology on N~(N~N) is not regular

摘要

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~((N~N)) formed in various cartesian closed categories of topological spaces. Moreover, we give an example of an effectively open subset of N~((N~N)) that violates regularity. The topological properties of N~((N~N)) 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.