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 show 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 X₁ ∨ X₂ = \(\frac{X_{1} ∪ X_{2}}{x_{1} ∼ x_{2}}\) has a categorical universal covering if and only if both X₁ and X₂ have categorical universal coverings.