Basic Topological Concepts and a Construction of Real Numbers in Alternative Set Theory

TL;DR

This paper develops basic topological concepts within Alternative Set Theory (AST), demonstrating their correspondence to classical concepts and establishing an isomorphism between the real numbers in AST and conventional real numbers.

Contribution

It introduces topological notions in AST and proves the isomorphism of real numbers in AST to classical real numbers, bridging the gap between AST and standard set theory.

Findings

Topological concepts in AST are formally defined.

Correspondence between AST and conventional topology is established.

Real numbers in AST are shown to be isomorphic to classical real numbers.

Abstract

Alternative set theory (AST) may be suitable for the ones who try to capture objects or phenomenons with some kind of indefiniteness of a border. While AST provides various notions for advanced mathematical studies, correspondence of them to that of conventional ones are not fully developed. This paper presents basic topological concepts in AST, and shows their correspondence with those of conventional ones, and isomorphicity of a system of real numbers in AST to that of conventional ones.