  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.
