Theory of representations (Q1073022)

An approach for a simple, general, and unified theory of effectivity on sets with cardinality not greater than that of the continuum is presented. A standard theory of effectivity on \({\mathbb{F}}=\{f:{\mathbb{N}}\to {\mathbb{N}}\}\) has been developed in a previous paper. By representations \(\delta\) :\({\mathbb{F}}\to M\) this theory is extended to other sets M. Topological and recursion theoretical properties of representations are studied, where the final topology of a representation plays an essential role. It is shown that for any separable \(T_ 0\)-space an (up to equivalence) unique admissible representation can be defined which reflects the topological properties correctly.