Admissible Representations of Limit Spaces

Abstract

We give a definition of admissible representations for (weak) limit spaces which allows to handle also non topological spaces in the framework of TTE (Type-2 Theory of Effectivity). Limit spaces and weak limit spaces spaces are generalizations of topological spaces. We prove that admissible representations

$$ \delta \mathfrak{X},\delta \mathfrak{Y}, $$

of weak limit spaces

$$ \mathfrak{X},\mathfrak{Y} $$

have the desirable property that every partial function f between them is continuously realizable with respect to

$$ \delta \mathfrak{X},\delta \mathfrak{Y}, $$

iff f is sequentially continuous. Furthermore, we characterize the class of the spaces having an admissible representation. The category of these spaces (equipped with the total sequential continuous functions as morphisms) turns out to be bicartesian-closed. It contains all countably-based T _o-spaces. Thus, a reasonable computability theory is possible on important non countably-based topological spaces as well as on non topological spaces.