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 π1(X,x) is uncountable or X
Nearby in the stack
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
X1∨X2=x1∼x2X1∪X2
has a categorical universal covering if and only if both