Abstract. The relationship between computability and continuity is studied. Computability over an arbitrary initial basis of data types and functions (a base) is considered using McCarthy recursive schemata and strongly typed operators of finite types. In this case, computable operators are proved to be strongly continuous in the Baire sense: for parameter functions with any value of the other arguments, it is possible to find a finite collection of their values that uniquely determines the result. Based on relative computability, an approach to constructive topology is developed within which the pointwise approach (an element is a fundamental sequence of neighborhoods) and the approximation approach of abstract topology (a function over topological spaces is a neighborhood relation) are equivalent. The concept of B-spaces is formulated, allowing one to constructivize separable spaces with a countable base of neighborhoods. The continuity of pointwise constructive functions of B-spaces and their transformability into neighborhood relations are proved. The equivalence of the concepts of computability relative to a certain base and continuity is established. The concept of a relatively constructive function is formulated: such a function transforms each element into an element constructible relative to its argument and a fixed base. Its equivalence to the concept of a countably continuous function formed by the union of a countable family of functions continuous on subspaces is established. Since any separable space with a countable base can be described as a B-space, this result contains no constructive restrictions. The connection between the proposed approach and other approaches to constructive topology is discussed.
Keywords: relative computability, separability, countable base, pointwise spaces, abstract topology, approximation approach, Baire continuity, continuity on subspaces.
Acknowledgments. This work was carried out within project no. 125021302067-9 of Ailamazyan Program Systems Institute, the Russian Academy of Sciences.
Continuity as Computability