Class of extensions of real field and their topological properties

TL;DR

This paper explores special classes of real field extensions, analyzing their topological properties and potential applications in measure theory, particularly in relation to sets of zero Lebesgue measure.

Contribution

It introduces new classes of real field extensions and studies their topological characteristics, linking algebraic structure with topological and measure-theoretic properties.

Findings

Connected extensions are not closed under addition and multiplication.

Non-connected extensions are linearly ordered fields.

Potential applications in measure theory and zero Lebesgue measure sets.

Abstract

Proper classes of extensions of real field was defined and topological properties of these extensions were studied. These extensions can be connected, in this case such set is not closed under binary operations (addition and multiplication), and not connected, in this case this extension is linearly ordered field. In the future these constructions can be applied to building measure that "feels" set of zero Lebesgue measure.