Iosif Petrakis - Bishop Spaces and the Problem of Constructivizing General Topology

Explore the constructive approach to general topology through Bishop spaces in this lecture from the Hausdorff Trimester Program on Constructive Mathematics. Delve into how Bishop spaces overcome the obstacles in constructivizing classical topological spaces, as identified by Errett Bishop. Examine the function-theoretic nature of Bishop spaces and how Bishop morphisms generalize uniform continuity. Discover key results on pseudocompact Bishop spaces and their potential as a constructive alternative to classical C*(X) theory. Gain insights into the connections between Bishop spaces and other constructive notions of space, including uniform spaces with pseudometric uniformities and C-spaces. Learn about the fundamental concepts, theorems, and applications of Bishop spaces in constructive mathematics, supported by references to seminal works in the field.