On the Relationship between Filter Spaces and Weak Limit Spaces

Countably based filter spaces have been suggested in the 1970's as a model for recursion theory on higher types. Weak limit spaces with a countable limit base are known to be the class of spaces which can be handled by the Type-2 Model of Effectivity (TTE). We prove that the category of countably based filter spaces is equivalent to the category of weak limit spaces with a countable limit base. As a consequence we obtain that filter spaces form yet another category from which the category QCB of quotients of countably based topological spaces inherits its cartesian closed structure.