The comparison of topologies related to various concepts of generalized covering spaces

The most common construction of a generalized covering space is that of a topologized (appropriate version of a) path space X ˜ . There have been three suggested topologies on it, each with its advantages and disadvantages. They are called the Whisker topology, the Lasso topology and the quotient of the compact open topology. In this paper we study the relationship between these topologies. The main result consists of an example demonstrating that the Lasso topology is not finer that the compact open topology. sults also apply to the topology of the fundamental group which appears naturally as a subspace of X ˜ .