"Eminent domain" is the right of a city or State to take private propety for the good of the public use. Other countries use different terms for the same process, see: http://en.wikipedia.org/wiki/Eminent_domain
Annexation involves a territory being absorbed into a larger geo-political area, see:
http://en.wikipedia.org/wiki/Annexation
The area surrounding a city is often referred to as its metropolitan area or its suburbs.
The area around a city is known as its suburbs or metropolitan area. This includes the nearby towns, neighborhoods, and countryside that are connected to the city for economic or social reasons.
The core of an urban area is typically the central business district (CBD) where major commercial, cultural, and governmental activities are concentrated. It is characterized by high population density, tall buildings, and diverse land uses such as retail, office space, and entertainment venues. The core often serves as the economic and social hub of the city.
Metropolitan area
hinterland
A city-state.
The center of a city is often called the downtown or city center. It is typically the commercial, cultural, and historical heart of the city where most of the activity and major landmarks are located.
There is little room for housing in the Vatican. Most of the area is taken up by existing buildings and the Papal Gardens. People cannot simply decide to move to Vatican City. They must be employed there and request housing when it is available.
It might be an urban agglomeration, a conurbation or a metropolitan area.
It takes up 32,000,187 sq feet
The area surrounding a city is typically referred to as its "metropolitan area" or "city region." This region can include suburbs, smaller towns, and rural areas that are interconnected with the city in terms of social, economic, and cultural ties.
It is the 'City of London', usually just called 'The City' or 'The Square Mile'.