Hi: Is there a relation between k-connectedness ( meaning 1st, 2nd,..,k-th fundamental groups are trivial.) and having the homotopy-lifting property.? . This may be vaguely-related to being able to extend global sections from the j-th skeleton, to the (j+1)-st skeleton (j<=k, obviously), but I am not sure. How about k-connectedness for a pair (A,X) ( A a subspace of X). Thanks.