  We modify the global Skorokhod topology, on the space of cadlag paths, by
localising with respect to space variable, in order to include the eventual
explosions. The tightness of families of probability measures on the paths
space endowed with this local Skorokhod topology is studied and a
characterization of Aldous type is obtained. The local and global Skorokhod
topologies are compared by using a time change transformation. A number of
results in the paper should play an important role when studying L\'evy-type
processes with unbounded coefficients by martingale problem approach.
