On the Existence of Categorical Universal Coverings

TL;DR

This paper investigates conditions for the existence of categorical universal coverings in topological spaces, providing a generalized Shelah Theorem and criteria for unions of spaces, with implications for fundamental groups.

Contribution

It introduces necessary and sufficient conditions for categorical universal coverings and extends Shelah's Theorem to first countable Peano spaces.

Findings

A first countable Peano space has a categorical universal covering or an uncountable fundamental group.

The union of two spaces has a categorical universal covering if and only if both do.

Generalized Shelah Theorem applies to Peano continua and their fundamental groups.

Abstract

In this paper, we study necessary and sufficient conditions for the existence of categorical universal coverings using open covers of a given space $X$. As some applications, first we present a generalized version of the Shelah Theorem (Mycielski's conjecture: If $X$ is a Peano continuum, then $\pi_1(X,x)$ is uncountable or $X$ has a simply connected universal covering) which states that a first countable Peano space has a categorical universal covering or has an uncountable fundamental group. Second, we prove that the one point union $X_1\vee X_2=\frac{{X_1}\cup {X_2}}{{x_1}\sim {x_2}}$ has a categorical universal covering if and only if both $X_1$ and $X_2$ have categorical universal coverings.