A Computable Version of Dini's Theorem for Topological Spaces

摘要

By Dini's theorem on a compact metric space K any increasing sequence (g_i)_(i∈N) of real-valued continuous functions converging pointwise to a continuous function f converges uniformly. In this article we prove a fully computable version of a generalization: a modulus of uniform convergence can be computed from a quasi-compact subset K of a computable To-space with computable intersection, from an increasing sequence of lower semi-continuous real-valued functions on K and from an upper semi-continuous function to which the sequence converges. For formulating and proving we apply the representation approach to Computable Analysis (TTE). In particular, for the spaces of quasi-compact subsets and of the partial semi-continuous functions we use natural multi-representations. Moreover, the operator computing a modulus of convergence is multi-valued.