Extended admissibility. (Q1607303)

We give a new definition of admissible representations which allows to handle also non countably-based topological spaces in the framework of Type-2 Theory of Effectivity. We prove that admissible representations \(\delta_{X},\delta_{Y}\) of topological spaces \(X,Y\) have the desirable property that every partial function \(f\subseteq X\rightarrow Y\) is continuously realizable with respect to \(\delta_{X},\delta_{Y}\) if and only if \(f\) is sequentially continuous. Furthermore, we characterize the class of the spaces having an admissible representation. Many interesting operators creating new topological spaces from old ones are shown to preserve the property of having an admissible representation. In particular, the class of sequential spaces with admissible representations turns out to be cartesian-closed. Thus, a reasonable computability theory is possible on important non countably based spaces.