For the stability of generalized games' Nash equilibria, current researches are investigated by uniform metric topology of feasible strategy correspondence. However, this paper presents a weaker metric and uses Hausdorff distance of graph of feasible strategy correspondence to study the stability of Nash equilibria. Under weaker graph topology, we prove that the space of generalized games is complete, and Nash equilibrium correspondence is upper semi-continuous and compact-valued. These lead to generic stability of generalized games' Nash equilibria, i.e., in the sense of Baire category, most of generalized games are essential.