Covering Maps with respect to Topologies on the Fundamental Group

TL;DR

This paper generalizes classical covering maps by introducing a new concept of ^{ au}-covering maps, which unify various types of coverings through topologies on the fundamental group, and compares their properties.

Contribution

It introduces a new generalized covering map concept ^{ au}-covering, unifying classical, semi, generalized coverings, and fibrations under a topological framework on the fundamental group.

Findings

^{ au}-covering maps encompass classical and generalized coverings.

Comparison of ^{ au}-coverings for different topologies on (X).

Properties of ^{ au}-coverings are established and analyzed.

Abstract

In this paper, using the classical covering theory, we introduce a generalization of covering maps of a space $X$ with respect to a topology $\tau$ on the fundamental group of $X$. We show that the famous notions, covering, semicovering, generalized covering and fibration maps are of special cases of this new notion $\pi_1^{\tau}$-covering map. Moreover, among presenting some properties for this new notion, we compare $\pi_1^{\tau}$-covering maps of a space $X$ for several famous topologies on the fundamental group of $X$.