Graph topologies for nonlinear operators which admit coprime factorizations are defined w. r. t. a gain function notion of stability in a general normed signal space setting. Several metrics are also defined and their relationship to the graph topologies are examined. In particular, relationships between nonlinear generalizations of the gap and graph metrics, Georgiou- type formulae, and the graph topologies are established. Closed loop robustness results are given w. r. t. the graph topology, where the role of a coercivity condition on the nominal plant is emphasized.