Nonstandard Topology without Logic, Ultrafilters as infinitesimal points in topological space

TL;DR

This paper proposes using ultrafilters as a simple way to describe infinitesimal points in topological spaces, aiming to simplify nonstandard analysis concepts.

Contribution

It introduces ultrafilters as a novel tool to represent infinitesimal points, providing a new perspective in topological and nonstandard analysis.

Findings

Ultrafilters can effectively model infinitesimal points.

Topological concepts are restated using ultrafilters.

Simplifies the understanding of nonstandard analysis.

Abstract

Nonstandard analysis is very complex, so finding a simple description of infinitesimal points will be useful. In this paper, ultrafilters as infinitesimal points in a topological space will be proposed, and some topological concepts is restated by this tools.