GENERAL TYPE‐STRUCTURES OF CONTINUOUS AND COUNTABLE FUNCTIONALS

KLEEXE [12] and KREISEL [13] independently introduced the countable or continuous functionals. KLEENE defined the countable functionals as a subclass of the total functionals while KRETSEL defined the continuous functionals as equivalence-classes of functions a : N -+ N. In recent expositions the continuous or countable functionals have been regarded as the elements of a certain type-structure (ct(h))k.N. This is closer to #REISEL'S approach than to KLEENE'S. There are several characterizations of the countable functionals, BERGSTRA [a] gave a recursiontheoretic characterization and HPLAXD [9] gave characterizations using topological spaces, filter spaces and limit spaces. Alternative characterizations are also given by ERSHOV [4].