Exact computation over topological spaces

TL;DR

This paper presents Natural Topology (NToP), a framework for exact computation on topological spaces, offering a new representation of real numbers and continuous functions that aligns with recent expert recommendations.

Contribution

It introduces the NToP framework, providing a new, efficient way to represent and compute with real numbers and continuous functions on natural spaces, extending existing theory.

Findings

NToP matches recent expert recommendations for exact real computation

Derived strong new results on representing continuous real-valued functions

Expanded results to a broad class of continuous functions between natural spaces

Abstract

We give an exposition of Natural Topology (NToP), which highlights its advantages for exact computation. The NToP-definition of the real numbers (and continuous real functions) matches recent expert recommendations for exact real computation (see [Bauer&Kavkler2008] and [Bauer&Kavkler2009]). We retrieve existing theory and derive strong new results on the efficient representation of continuous real-valued functions defined on a general class of topological spaces (called natural spaces). We then expand these results to a large class of continuous functions between natural spaces.