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 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 proper filter spaces is equivalent to the category of countably based weak limit spaces. This result implies that filter spaces form yet another category from which the category of qcb-spaces inherits its cartesian closed structure. Moreover, we compare the aforementioned categories to other categories of spaces relevant to computability theory.