CARTESIAN CLOSEDNESS, EXPONENTIALITY, AND FINAL HULLS IN PSEUDOTOPOLOGICAL SPACES

Abstract

We show that each non-trivial epireflective subcategory of the topological or pretopological spaces fails to be cartesian closed. Motivated by this “negative” result, we consider the supercategory of pseudotopological spaces and obtain: An epireflective subcategory of the pseudotopological spaces which contains a finite non-indiscrete space is cartesian closed iff it is closed with respect to powers in the pseudotopological spaces. Here the density property that every pseudotopological space is a final epi-sink of free ultraspaces is essential.