According to Arizona law, what defines counties in the state?

Counties in Arizona are defined as specific political subdivisions of the state that have been established by law. They serve as a way to administer government services at a local level and are recognized entities within the structure of Arizona's governance. Each county operates with its own local form of government and is responsible for various functions such as public safety, health services, and infrastructure maintenance.

The other options do not accurately capture the formal role and definition of counties. Incorporated territories and independent districts refer to different types of governmental entities and administrative organizations, which do not align with the established structure of counties. Administrative regions could refer to various organizational layouts for districts or services but do not specifically denote the legal and governmental framework that defines counties in Arizona. Thus, the choice that identifies counties as political subdivisions of the state is the most accurate representation of their definition within Arizona law.