Algebraic Topology Without Open Sets: A Net Approach to Homotopy Theory in Limit Spaces

TL;DR

This paper introduces a net-based approach to convergence spaces, simplifying continuous convergence descriptions and extending homotopy theory concepts like the fundamental groupoid and Seifert-van Kampen theorem to limit spaces.

Contribution

It develops a net-theoretic framework for convergence spaces, enabling algebraic topology tools to be applied without open sets.

Findings

Defined the fundamental groupoid for limit spaces

Proved the Seifert-van Kampen theorem in this context

Simplified the description of continuous convergence

Abstract

Convergence spaces are a generalization of topological spaces. The category of convergence spaces is well-suited for Algebraic Topology, one of the reasons is the existence of exponential objects provided by continuous convergence. In this work, we use a net-theoretic approach to convergence spaces. The goal is to simplify the description of continuous convergence and apply it to problems related to homotopy theory. We present methods to develop the basis of homotopy theory in limit spaces, define the fundamental groupoid, and prove the groupoid version of the Seifert-van Kampen Theorem for limit spaces.