Ohio Rev. Code § 3311.02: City school district defined.

The territory within the corporate limits of each city, excluding the territory detached therefrom for school purposes and including the territory attached thereto for school purposes, constitutes a city school district. When a city is reduced to a village, the city school district shall thereupon become a local school district, except that when a city is reduced to a village but its city school district includes within its boundaries all or part of two or more municipal corporations, the aggregate population of which totals five thousand or more as determined by the preceding federal census, it may, by a majority vote of the school board, remain a city school district. An exempted village school district which includes within its boundaries all or part of two or more municipal corporations, the aggregate population of which totals five thousand or more as determined by the preceding federal census, may, with the approval of the state board of education, become a city school district.