Domain Representable Spaces Defined by Strictly Positive Induction

TL;DR

This paper investigates how strictly positive inductive definitions can be used to construct and analyze $qcb_{0}$ spaces through domain representations, establishing a canonical fixed point for such operations.

Contribution

It demonstrates the existence of a canonical fixed point for strictly positive operations on $qcb_{0}$ spaces using domain representations, advancing the theory of domain-based topological spaces.

Findings

Existence of canonical fixed points for strictly positive operations on $qcb_{0}$ spaces.

Extension of domain representation techniques to $qcb_{0}$ spaces.

Framework for constructing $qcb_{0}$ spaces via positive induction.

Abstract

Recursive domain equations have natural solutions. In particular there are domains defined by strictly positive induction. The class of countably based domains gives a computability theory for possibly non-countably based topological spaces. A $ qcb_{0} $ space is a topological space characterized by its strong representability over domains. In this paper, we study strictly positive inductive definitions for $ qcb_{0} $ spaces by means of domain representations, i.e. we show that there exists a canonical fixed point of every strictly positive operation on $qcb_{0} $ spaces.