# Direct spectra of Bishop spaces and their limits

## TL;DR

This paper develops the theory of direct and inverse limits of Bishop spaces within Bishop set theory, providing constructive proofs of fundamental theorems and duality principles in Bishop's constructive mathematics.

## Contribution

It introduces the notions of spectra and limits of Bishop spaces and proves key theorems about their properties within an extended constructive framework.

## Key findings

- Established the existence of direct limits of Bishop spaces.
- Proved the fundamental theorems on inverse limits.
- Demonstrated a duality principle between direct and inverse spectra.

## Abstract

We apply fundamental notions of Bishop set theory (BST), an informal theory that complements Bishop's theory of sets, to the theory of Bishop spaces, a function-theoretic approach to constructive topology. Within BST we develop the notions of a direct family of sets, of a direct spectrum of Bishop spaces, of the direct limit of a direct spectrum of Bishop spaces, and of the inverse limit of a contravariant direct spectrum of Bishop spaces. Within the extension of Bishop's informal system of constructive mathematics BISH with inductive definitions with rules of countably many premises, we prove the fundamental theorems on the direct and inverse limits of spectra of Bishop spaces and the duality principle between them.