Natural Topology

TL;DR

This paper introduces natural topology, a framework combining pointwise and pointfree notions in Bishop-style constructive mathematics, with applications to real-world phenomena, physics, and computational representations of real numbers.

Contribution

It develops a comprehensive natural topology framework linking intuitionistic concepts with practical and computational applications, including examples like Hawk-Eye and real-number representations.

Findings

Constructive metrizability of star-finitary spaces

Examples of natural topology in applied mathematics and physics

Comparison of natural topology with classical and formal topologies

Abstract

We give a theoretical and applicable framework for dealing with real-world phenomena. Joining pointwise and pointfree notions in BISH, natural topology gives a faithful idea of important concepts and results in intuitionism. Natural topology is well-suited for practical and computational purposes. We give several examples relevant for applied mathematics, such as the decision-support system Hawk-Eye (used in professional tennis), and various real-number representations. We compare CLASS, INT, RUSS, BISH and formal topology. There are links with physics, regarding the topological character of our physical universe. Translation of intuitionistic results to BISH is facilitated by our framework, using transfinite countable-ordinal induction in Brouwer's style. We study quotients of Baire space, and obtain constructive metrizability of star-finitary spaces. Silva spaces arise as example of non-metrizable natural spaces. Finally we discuss the role of Church's Thesis in physics.