On the Existence of Categorical Universal Coverings

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.

arXiv comment13 pages, A revised version of "All Categorical Universal Coverings Are Spanier Spaces"

Cited by in corpus (1)

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