  Local properties of the fundamental group of a path-connected topological
space can pose obstructions to the applicability of covering space theory. A
generalized covering map is a generalization of the classical notion of
covering map defined in terms of unique lifting properties. The existence of
generalized covering maps depends entirely on the verification of the unique
path lifting property for a standard covering construction. Given any
path-connected metric space $X$, and a subgroup $H\leq\pi_1(X,x_0)$, we
characterize the unique path lifting property relative to $H$ in terms of a new
closure operator on the $\pi_1$-subgroup lattice that is induced by maps from a
fixed "test" domain into $X$. Using this test map framework, we develop a
unified approach to comparing the existence of generalized coverings with a
number of related properties.
